NCERT Solutions for Class 11 Maths Chapter 10
Straight Lines Class 11
Chapter 10 Straight Lines Exercise 10.1, 10.2, 10.3, miscellaneous, Solutions
Exercise 10.1 : Solutions of Questions on Page Number : 211
Draw a quadrilateral in the Cartesian plane, whose vertices are (-4, 5), (0, 7), (5, -5) and (-4, -2). Also, find its area.
Let ABCD be the given quadrilateral with vertices A (β4, 5), B (0, 7), C (5, β5), and D (β4, β2).
To find the area of quadrilateral ABCD, we draw one diagonal, say AC. Accordingly, area (ABCD) = area (ΔABC) + area (ΔACD)
We know that the area of a triangle whose vertices are (x1, y1), (x2, y2), and (x3, y3) is
Therefore, area of ΔABC
Thus, area (ABCD)
The base of an equilateral triangle with side 2a lies along they y-axis such that the mid point of the base is at the origin. Find vertices of the triangle.
Let ABC be the given equilateral triangle with side 2a. Accordingly, AB = BC = CA = 2a
Assume that base BC lies along the y-axis such that the mid-point of BC is at the origin. i.e., BO = OC = a, where O is the origin.
Now, it is clear that the coordinates of point C are (0, a), while the coordinates of point B are (0, βa).
It is known that the line joining a vertex of an equilateral triangle with the mid-point of its opposite side is perpendicular.
Hence, vertex A lies on the y-axis.
On applying Pythagoras theorem to ΔAOC, we obtain (AC)2 = (OA)2 + (OC)2
⇒ (2a)2 = (OA)2 + a2
⇒ 4a2 β a2 = (OA)2
⇒ (OA)2 = 3a2
⇒ OA =
∴Coordinates of point A =
Thus, the vertices of the given equilateral triangle are (0, a), (0, βa), and or (0, a), (0, βa), and
Find the distance between and when: (i) PQ is parallel to the y-axis, (ii) PQ is parallel to thex-axis.
The given points are and .
(i) When PQ is parallel to the y-axis, x1 = x2.
(ii) When PQ is parallel to the x-axis, y1 = y2.
Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).
On squaring both sides, we obtain
a2 β 14a + 85 = a2 β 6a + 25
⇒ β14a + 6a = 25 β 85
Thus, the required point on the x-axis is .
Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P (0, -4) and B (8, 0).
The coordinates of the mid-point of the line segment joining the points
P (0, β4) and B (8, 0) are
It is known that the slope (m) of a non-vertical line passing through the points (x1, y1) and (x2, y2) is given by .
Therefore, the slope of the line passing through (0, 0) and (4, β2) is
.
Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (-1, -1) are the vertices of a right angled triangle.
The vertices of the given triangle are A (4, 4), B (3, 5), and C (β1, β1).
It is known that the slope (m) of a non-vertical line passing through the points (x1, y1) and (x2, y2) is given by .
∴Slope of AB (m1) Slope of BC (m2) Slope of CA (m3)
It is observed that m1m3 = β1
This shows that line segments AB and CA are perpendicular to each other i.e., the given triangle is right-angled at A (4, 4).
Find the slope of the line, which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.
If a line makes an angle of 30° with the positive direction of the y-axis measured anticlockwise, then the angle made by the line with the positive direction of the x-axis measured anticlockwise is 90° + 30° = 120°.
Thus, the slope of the given line is tan 120° = tan (180° β 60°) = βtan 60°
Find the value of x for which the points (x, -1), (2, 1) and (4, 5) are collinear.
If points A (x, β1), B (2, 1), and C (4, 5) are collinear, then Slope of AB = Slope of BC
Without using distance formula, show that points (-2, -1), (4, 0), (3, 3) and (-3, 2) are vertices of a parallelogram.
Slope of AB
Slope of CD =
⇒ Slope of AB = Slope of CD
⇒ AB and CD are parallel to each other.
Now, slope of BC = Slope of AD =
⇒ Slope of BC = Slope of AD
⇒ BC and AD are parallel to each other.
Therefore, both pairs of opposite sides of quadrilateral ABCD are parallel. Hence, ABCD is a parallelogram. Thus, points (β2, β1), (4, 0), (3, 3), and (β3, 2) are the vertices of a parallelogram.
Find the angle between the x-axis and the line joining the points (3, -1) and (4, -2).
The slope of the line joining the points (3, β1) and (4, β2) is Now, the inclination (θ) of the line joining the points (3, β1) and (4, β 2) is given by
tan θ= β1
⇒ θ = (90°+ 45°) = 135°
The slope of a line is double of the slope of another line. If tangent of the angle between them is , find the slopes of he lines.
Let be the slopes of the two given lines such that .
If m = β1, then the slopes of the lines are β1 and β2.
If m = , then the slopes of the lines are and β1.
If m = 1, then the slopes of the lines are 1 and 2.
If m = , then the slopes of the lines are .
A line passes through . If slope of the line is m, show that . Answer :
The slope of the line passing through is .
Hence,
If three point (h, 0), (a, b) and (0, k) lie on a line, show that .
If the points A (h, 0), B (a, b), and C (0, k) lie on a line, then Slope of AB = Slope of BC
Hence,
Since line AB passes through points A (1985, 92) and B (1995, 97), its slope is
Let y be the population in the year 2010. Then, according to the given graph, line AB must pass through point C (2010, y).
Thus, the slope of line AB is , while in the year 2010, the population will be 104.5 crores.
Exercise 10.2 : Solutions of Questions on Page Number : 219
Write the equations for the x and y-axes.
The y-coordinate of every point on the x-axis is 0. Therefore, the equation of the x-axis is y = 0.
The x-coordinate of every point on the y-axis is 0. Therefore, the equation of the y-axis is x = 0.
Find the equation of the line which passes through the point (β4, 3) with slope . Answer :
We know that the equation of the line passing through point , whose slope is m, is . Thus, the equation of the line passing through point (β4, 3), whose slope is , is
Find the equation of the line which passes though (0, 0) with slope m.
We know that the equation of the line passing through point , whose slope is m, is .
Thus, the equation of the line passing through point (0, 0), whose slope is m,is (y β 0) = m(x β 0)
Find the equation of the line which passes though and is inclined with the x-axis at an angle of 75°.
The slope of the line that inclines with the x-axis at an angle of 75° is
We know that the equation of the line passing through point , whose slope is m, is . Thus, if a line passes though and inclines with the x-axis at an angle of 75°, then the equation of the line is given as
Find the equation of the line which intersects the x-axis at a distance of 3 units to the left of origin with slope
Answer :
It is known that if a line with slope m makes x-intercept d, then the equation of the line is given as
y = m(x - d)
For the line intersecting the x-axis at a distance of 3 units to the left of the origin, d = -3. The slope of the line is given as m = -2
Thus, the required equation of the given line is
y = -2 [x - (-3)]
y = -2x - 6
Find the equation of the line which intersects the y-axis at a distance of 2 units above the origin and makes an angle of 30° with the positive direction of the x-axis.
It is known that if a line with slope m makes y-intercept c, then the equation of the line is given as
y = mx + c
Here, c = 2 and m = tan 30° .
Thus, the required equation of the given line is
Find the equation of the line which passes through the points (-1, 1) and (2, -4).
It is known that the equation of the line passing through points (x1, y1) and (x2, y2) is .
Therefore, the equation of the line passing through the points (β1, 1) and (2, β4) is
Find the equation of the line which is at a perpendicular distance of 5 units from the origin and the angle made by the perpendicular with the positive x-axis is 30°
If p is the length of the normal from the origin to a line and Γβ°is the angle made by the normal with the positive direction of the x-axis, then the equation of the line is given by xcos Γβ° + y sin Γβ°= p.
Here, p = 5 units and Γβ° = 30°
Thus, the required equation of the given line is
x cos 30° + y sin 30° = 5
The vertices of ΔPQR are P (2, 1), Q (-2, 3) and R (4, 5). Find equation of the median through the vertex R.
It is given that the vertices of ΔPQR are P (2, 1), Q (β2, 3), and R (4, 5). Let RL be the median through vertex R.
Accordingly, L is the mid-point of PQ.
By mid-point formula, the coordinates of point L are given by
It is known that the equation of the line passing through points (x1, y1) and (x2, y2) is . Therefore, the equation of RL can be determined by substituting (x1, y1) = (4, 5) and (x2, y2) = (0, 2).
Thus, the required equation of the median through vertex R is .
Find the equation of the line passing through (-3, 5) and perpendicular to the line through the points (2, 5) and (-3, 6).
The slope of the line joining the points (2, 5) and (β3, 6) is
We know that two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.
A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1:n. Find the equation of the line.
The slope of the line joining the points (1, 0) and (2, 3) is
We know that two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.
Therefore, slope of the line that is perpendicular to the line joining the points (1, 0) and (2, 3) Now, the equation of the line passing through and whose slope is is given by
Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).
Here, a and b are the intercepts on x and y axes respectively.
It is given that the line cuts off equal intercepts on both the axes. This means that a = b. Accordingly, equation (i) reduces to
Since the given line passes through point (2, 3), equation (ii) reduces to 2 + 3 = a ⇒ a = 5
On substituting the value of a in equation (ii), we obtain
Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.
Here, a and b are the intercepts on x and y axes respectively.
It is given thata + b = 9 ⇒ b = 9 β a (ii) From equations (i) and (ii), we obtain
If a = 6 and b = 9 β 6 = 3, then the equation of the line is
Find equation of the line through the point (0, 2) making an angle with the positive x-axis. Also, find the equation of line parallel to it and crossing the y-axis at a distance of 2 units below the origin.
The slope of the line making an angle with the positive x-axis is
The slope of line parallel to line is .
It is given that the line parallel to line crosses the y-axis 2 units below the origin i.e., it passes through point (0, β2).
The perpendicular from the origin to a line meets it at the point (- 2, 9), find the equation of the line.
Now, the equation of the line passing through point (β2, 9) and having a slope m2 is
The length L (in centimetre) of a copper rod is a linear function of its Celsius temperature C. In an experiment, if L = 124.942 when C = 20 and L = 125.134 when C = 110, express L in terms of C.
It is given that when C = 20, the value of L is 124.942, whereas when C = 110, the value of L is 125.134. Accordingly, points (20, 124.942) and (110, 125.134) satisfy the linear relation between L and C.
Now, assuming C along the x-axis and L along the y-axis, we have two points i.e., (20, 124.942) and (110, 125.134) in the XY plane.
Therefore, the linear relation between L and C is the equation of the line passing through points (20, 124.942) and (110, 125.134).
The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear relationship between selling price and demand, how many litres could he sell weekly at Rs 17/litre?
The relationship between selling price and demand is linear.
Assuming selling price per litre along the x-axis and demand along the y-axis, we have two points i.e., (14, 980) and (16, 1220) in the XY plane that satisfy the linear relationship between selling price and demand.
Thus, the owner of the milk store could sell 1340 litres of milk weekly at Rs 17/litre.
P (a, b) is the mid-point of a line segment between axes. Show that equation of the line is Answer :
Let the coordinates of A and B be (0, y) and (x, 0) respectively. Since P (a, b) is the mid-point of AB,
Thus, the respective coordinates of A and B are (0, 2b) and (2a, 0). The equation of the line passing through points (0, 2b) and (2a, 0) is
On dividing both sides by ab, we obtain
Point R (h, k) divides a line segment between the axes in the ratio 1:2. Find equation of the line.
Let the respective coordinates of A and B be (x, 0) and (0, y).
Therefore, the respective coordinates of A and B are and (0, 3k).
Now, the equation of line AB passing through points and (0, 3k) is
By using the concept of equation of a line, prove that the three points (3, 0), (-2, -2) and (8, 2) are collinear.
In order to show that points (3, 0), (β2, β2), and (8, 2) are collinear, it suffices to show that the line passing through points (3, 0) and (β2, β2) also passes through point (8, 2).
It is observed that at x = 8 and y = 2,
L.H.S. = 2 Χ 8 β 5 Χ 2 = 16 β 10 = 6 = R.H.S.
Exercise 10.3 : Solutions of Questions on Page Number : 227
Reduce the following equations into slope-intercept form and find their slopes and the y-intercepts. (i) x + 7y = 0 (ii) 6x + 3y - 5 = 0 (iii) y = 0
(i) The given equation is x + 7y = 0.
This equation is of the form y = mx + c, where .
Therefore, equation (1) is in the slope-intercept form, where the slope and the y-intercept are and 0 respectively.
(ii) The given equation is 6x + 3y β 5 = 0. It can be written as
Therefore, equation (2) is in the slope-intercept form, where the slope and the y-intercept areβ2 and respectively.
(iii) The given equation is y = 0.
It can be written as
y = 0.x + 0 (3)
This equation is of the form y = mx + c, where m = 0 and c = 0.
Reduce the following equations into intercept form and find their intercepts on the axes. (i) 3x + 2y - 12 = 0 (ii) 4x - 3y = 6 (iii) 3y + 2 = 0.
(i) The given equation is 3x + 2y β 12 = 0. It can be written as
This equation is of the form , where a = 4 and b = 6.
Therefore, equation (1) is in the intercept form, where the intercepts on the x and y axes are 4 and 6 respectively.
(ii) The given equation is 4x β 3y = 6. It can be written as
This equation is of the form , where a = and b = β2.
Therefore, equation (2) is in the intercept form, where the intercepts on the x and y axes are and β2 respectively.
(iii) The given equation is 3y + 2 = 0. It can be written as
This equation is of the form , where a = 0 and b = .
Reduce the following equations into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.
Answer :
(i)
On dividing both sides by , we obtain
Equation (1) is in the normal form.
On comparing equation (1) with the normal form of equation of line
xcos Γβ°+ y sin Γβ°= p, we obtain Γβ°= 120°and p = 4.
Thus, the perpendicular distance of the line from the origin is 4, while the angle between the perpendicular and the positive x-axis is 120°.
(ii) The given equation is y β 2 = 0. Itcan be reduced as 0.x+ 1.y = 2
On dividing both sides by , we obtain 0.x+ 1.y = 2
⇒ x cos 90°+ y sin 90°= 2 (1) Equation (1) is in the normal form.
On comparing equation (1) with the normal form of equation of line
xcos Γβ°+ y sin Γβ°= p, we obtain Γβ°= 90°and p = 2.
Thus, the perpendicular distance of the line from the origin is 2, while the angle between the perpendicular and the positive x-axis is 90°.
(iii) The given equation is x β y = 4. Itcan be reduced as 1.x+ (β1) y = 4
On dividing both sides by , we obtain
Equation (1) is in the normal form.
On comparing equation (1) with the normal form of equation of line
xcos Γβ°+ y sin Γβ°= p, we obtain Γβ°= 315°and .
Find the distance of the point (-1, 1) from the line 12(x + 6) = 5(y - 2).
The given equation of the line is 12(x + 6) = 5(y β 2).
⇒ 12x + 72 = 5y β 10
⇒12x β 5y + 82 = 0 (1)
On comparing equation (1) with general equation of line Ax + By + C = 0, we obtain A = 12, B = β5, and C = 82. It is known that the perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by
.
The given point is (x1, y1) = (β1, 1).
Find the points on the x-axis, whose distances from the line are 4 units. Answer :
On comparing equation (1) with general equation of line Ax + By + C = 0, we obtain A = 4, B = 3, and C = β12. Let (a, 0) be the point on the x-axis whose distance from the given line is 4 units.
It is known that the perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by
.
Thus, the required points on the x-axis are (β2, 0) and (8, 0).
Find the distance between parallel lines (i) 15x + 8y - 34 = 0 and 15x + 8y + 31 = 0
(ii) l (x + y) + p = 0 and l (x + y) - r = 0
It is known that the distance (d) between parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0 is given by .
(i) The given parallel lines are 15x + 8y β 34 = 0 and 15x + 8y + 31 = 0. Here, A = 15, B = 8, C1 = β34, and C2 = 31.
Therefore, the distance between the parallel lines is
(ii) The given parallel lines are l (x + y) + p = 0 and l (x + y) β r = 0.
lx + ly + p = 0 and lx + ly β r = 0 Here, A = l, B = l, C1 = p, and C2 = βr.
Find equation of the line parallel to the line 3x - 4y + 2 = 0 and passing through the point (-2, 3).
, which is of the form y = mx + c
∴ Slope of the given line
It is known that parallel lines have the same slope.
∴ Slope of the other line =
Find equation of the line perpendicular to the line x - 7y + 5 = 0 and having x intercept 3.
The given equation of line is .
, which is of the form y = mx + c
∴Slope of the given line
The slope of the line perpendicular to the line having a slope of is The equation of the line with slope β7 and x-intercept 3 is given by
y = m (x β d)
⇒ y = β7 (x β 3)
⇒ y = β7x + 21
Find angles between the lines Answer :
The slope of line (1) is , while the slope of line (2) is . The acute angle i.e., θ between the two lines is given by
The line through the points (h, 3) and (4, 1) intersects the line 7x - 9y - 19 = 0. at right angle. Find the value of h.
The slope of line 7x β 9y β 19 = 0 or is . It is given that the two lines are perpendicular.
Prove that the line through the point (x1, y1) and parallel to the line Ax + By + C = 0 is A (x -x1) + B (y - y1) = 0.
The slope of line Ax + By + C = 0 or is It is known that parallel lines have the same slope.
∴ Slope of the other line =
Hence, the line through point (x1, y1) and parallel to line Ax + By + C = 0 is A (x βx1) + B (y β y1) = 0
Two lines passing through the point (2, 3) intersects each other at an angle of 60°. If slope of one line is 2, find equation of the other line.
It is given that the slope of the first line, m1 = 2. Let the slope of the other line be m2.
The angle between the two lines is 60°.
In this case, the equation of the other line is .
The equation of the line passing through point (2, 3) and having a slope of is
In this case, the equation of the other line is .
Find the equation of the right bisector of the line segment joining the points (3, 4) and (-1, 2).
The right bisector of a line segment bisects the line segment at 90°.
The end-points of the line segment are given as A (3, 4) and B (β1, 2).
Accordingly, mid-point of AB Slope of AB
∴Slope of the line perpendicular to AB =
The equation of the line passing through (1, 3) and having a slope of β2 is (y β 3) = β2 (x β 1)
y β 3 = β2x + 2 2x + y = 5
Find the coordinates of the foot of perpendicular from the point (-1, 3) to the line 3x - 4y - 16 = 0.
Let (a, b) be the coordinates of the foot of the perpendicular from the point (β1, 3) to the line 3x β 4y β 16 = 0.
Slope of the line joining (β1, 3) and (a, b), m1
Point (a, b) lies on line 3x β 4y = 16.
∴3a β 4b = 16 (2)
Thus, the required coordinates of the foot of the perpendicular are .
The perpendicular from the origin to the line y = mx + c meets it at the point (-1, 2). Find the values of m and c.
The given equation of line is y = mx + c.
It is given that the perpendicular from the origin meets the given line at (β1, 2). Therefore, the line joining the points (0, 0) and (β1, 2) is perpendicular to the given line.
∴Slope of the line joining (0, 0) and (β1, 2)
Since point (β1, 2) lies on the given line, it satisfies the equation y = mx + c.
If p and q are the lengths of perpendiculars from the origin to the lines x cos ΓΒΈ - y sin ΓΒΈ = k cos 2ΓΒΈ and x sec ΓΒΈ+ y cosec ΓΒΈ = k, respectively, prove that p2 + 4q2 = k2
The equations of given lines are
x cos θ β y sinθ = k cos 2θ (1)
x secθ + y cosec θ= k (2)
The perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by . On comparing equation (1) to the general equation of line i.e., Ax + By + C = 0, we obtain A = cosθ, B = βsinθ, and C = βk cos 2θ.
It is given that p is the length of the perpendicular from (0, 0) to line (1).
On comparing equation (2) to the general equation of line i.e., Ax + By + C = 0, we obtain A = secθ, B = cosecθ, and C = βk.
From (3) and (4), we have
In the triangle ABC with vertices A (2, 3), B (4, -1) and C (1, 2), find the equation and length of altitude from the vertex A.
The equation of the line passing through point (2, 3) and having a slope of 1 is (y β 3) = 1(x β 2)
⇒ x β y + 1 = 0
⇒ y β x = 1
Therefore, equation of the altitude from vertex A = y β x = 1. Length of AD = Length of the perpendicular from A (2, 3) to BC The equation of BC is
The perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by . On comparing equation (1) to the general equation of line Ax + By + C = 0, we obtain A = 1, B = 1, and C = β3.
∴Length of AD
If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that .
The perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by . On comparing equation (1) to the general equation of line Ax + By + C = 0, we obtain A = b, B = a, and C = βab.
On squaring both sides, we obtain
Exercise Miscellaneous : Solutions of Questions on Page Number : 233
Find the values of k for which the line is
(b) Parallel to the y-axis,
Answer :
The given equation of line is
(k β 3) x β (4 β k2) y + k2 β 7k + 6 = 0 (1)
(a) If the given line is parallel to the x-axis, then Slope of the given line = Slope of the x-axis The given line can be written as
(4 β k2) y = (k β 3) x + k2 β 7k + 6 = 0
, which is of the form y = mx + c.
Thus, if the given line is parallel to the x-axis, then the value of k is 3.
(b) If the given line is parallel to the y-axis, it is vertical. Hence, its slope will be undefined.
The slope of the given line is .
Now, is undefined at k2 = 4
k2 = 4
⇒ k = ±2
Thus, if the given line is parallel to the y-axis, then the value of k is ±2.
(c) If the given line is passing through the origin, then point (0, 0) satisfies the given equation of line.
Thus, if the given
line is passing through the origin, then the value of k is either 1 or 6.
Find the values of θand p, if the equation is the normal form of the line .
On dividing both sides by , we obtain
Since the values of sin θ and cos θ are negative, Thus, the respective values of θand p are and 1
Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and -6, respectively.
Let the intercepts cut by the given lines on the axes be a and b.
It is given that
a + b = 1 (1)
ab = β6 (2)
On solving equations (1) and (2), we obtain
a = 3 and b = β2 or a = β2 and b = 3
Case I: a = 3 and b = β2
In this case, the equation of the line is β2x + 3y + 6 = 0, i.e., 2x β 3y = 6.
Case II: a = β2 and b = 3
In this case, the equation of the line is 3x β 2y + 6 = 0, i.e., β3x + 2y = 6. Thus, the required equation of the lines are 2x β 3y = 6 and β3x + 2y = 6.
What are the points on the y-axis whose distance from the line is 4 units. Answer :
Let (0, b) be the point on the y-axis whose distance from line is 4 units. The given line can be written as 4x + 3y β 12 = 0 (1)
On comparing equation (1) to the general equation of line Ax + By + C = 0, we obtain A = 4, B = 3, and C = β12. It is known that the perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by
.
Therefore, if (0, b) is the point on the y-axis whose distance from line is 4 units, then:
Thus, the
required points are and .
Find the perpendicular distance from the origin to the line joining the points
It is known that the perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by
.
Therefore, the perpendicular distance (d) of the given line from point (x1, y1) = (0, 0) is
Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines x - 7y+ 5 = 0 and 3x + y = 0.
The equation of any line parallel to the y-axis is of the form
x = a (1)
The two given lines are x β 7y + 5 = 0 (2) 3x + y = 0 (3)
On solving equations (2) and (3), we obtain .
Therefore, is the point of intersection of lines (2) and (3).
Since line x = a passes through point , .
Thus, the required
equation of the line is .
Find the equation of a line drawn perpendicular to the line through the point, where it meets the y-axis.
The equation of the given line is .
This equation can also be written as 3x + 2y β 12 = 0
, which is of the form y = mx + c
∴Slope of the given line
∴Slope of line perpendicular to the given line Let the given line intersect the y-axis at (0, y).
On substituting x with 0 in the equation of the given line, we obtain
∴The given line intersects the y-axis at (0, 6).
Thus, the required equation of the line is .
Find the area of the triangle formed by the lines y -x = 0, x + y = 0 and x - k = 0.
The equations of the given lines are
y β x = 0 (1)
x + y = 0 (2)
x β k = 0 (3)
The point of intersection of lines (1) and (2) is given by
x = 0 and y = 0
The point of intersection of lines (2) and (3) is given by
x = k and y = βk
The point of intersection of lines (3) and (1) is given by
x = k and y = k
Thus, the vertices of the triangle formed by the three given lines are (0, 0), (k, βk), and (k, k). We know that the area of a triangle whose vertices are (x1, y1), (x2, y2), and (x3, y3) is
.
Find the value of p so that the three lines 3x + y - 2 = 0, px + 2y - 3 = 0 and 2x - y - 3 = 0 may intersect at one point.
The equations of the given lines are 3x + y - 2 = 0 (1)
px + 2y - 3 = 0 (2)
2x - y - 3 = 0 (3)
On solving equations (1) and (3), we obtain
x = 1 and y = -1
Since these three lines may intersect at one point, the point of intersection of lines (1) and (3) will also satisfy line (2).
p (1) + 2 (-1) - 3 = 0
p - 2 - 3 = 0
p = 5
Thus, the
required value of p is 5.
If three lines whose equations are concurrent, then show that
The equations of the given lines are
y = m1x + c1 (1) y = m2x + c2 (2) y = m3x + c3 (3)
On substituting this value of x in (1), we obtain
is the point of intersection of lines (1) and (2).
Hence,
Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line x -2y = 3.
Let the slope of the required line be m1.
The given line can be written as , which is of the form y = mx + c
∴Slope of the given line =
It is given that the angle between the required line and line x β 2y = 3 is 45°.
We know that if θisthe acute angle between lines l1 and l2 with slopes m1 and m2 respectively, then
.
Case I: m1 = 3
The equation of the line passing through (3, 2) and having a slope of 3 is:
y β 2 = 3 (x β 3) y β 2 = 3x β 9 3x β y = 7
Case II: m1 =
Thus, the equations of the lines are 3x β y = 7 and x + 3y = 9.
Find the equation of the line passing through the point of intersection of the lines 4x + 7y - 3 = 0 and 2x - 3y+ 1 = 0 that has equal intercepts on the axes.
On solving equations 4x + 7y β 3 = 0 and 2x β 3y + 1 = 0, we obtain . is the point of intersection of the two given lines.
∴ Equation (1) becomes Thus, the required equation of the line is .
Exercise Miscellaneousmiscellaneous : Solutions of Questions on Page Number : 234
Show that the equation of the line passing through the origin and making an angle θwith the line
Answer :
Let the equation of the line passing through the origin be y = m1x.
If this line makes an angle of θ with line y = mx + c, then angle θ is given by
Case II:
Therefore, the required
line is given by .
In what ratio, the line joining (-1, 1) and (5, 7) is divided by the line
x + y = 4?
The equation of the given line is
x + y β 4 = 0 (2)
The point of intersection of lines (1) and (2) is given by
x = 1 and y = 3
Let point (1, 3) divide the line segment joining (β1, 1) and (5, 7) in the ratio 1:k. Accordingly, by section formula,
Thus, the line joining the points (β1, 1) and (5, 7) is divided by line
x + y = 4 in the ratio
1:2.
Find the distance of the line 4x + 7y + 5 = 0 from the point (1, 2) along the line 2x - y = 0.
The given lines are 2x β y = 0 (1)
4x + 7y + 5 = 0 (2)
A (1, 2) is a point on line (1).
On solving equations (1) and (2), we obtain .
∴Coordinates of point B are .
By using distance formula, the distance between points A and B can be obtained as
Thus, the
required distance is .
Find the direction in which a straight line must be drawn through the point (-1, 2) so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point.
Let y = mx + c be the line through point (β1, 2). Accordingly, 2 = m (β1) + c.
⇒ 2 = βm + c
⇒ c = m + 2
∴ y = mx + m + 2 (1)
The given line is
x + y = 4 (2)
On solving equations (1) and (2), we obtain
is the point of intersection of lines (1) and (2).
Thus, the slope of the required line must be zero i.e., the line must be parallel to the x-axis.
Answer :
Let A(1,3) and B(-4,1) be the coordinates of the end points of the hypotenuse.
Now, plotting the line segment joining the points A(1,3) and B(-4,1) on the coordinate plane, we will get two right triangles with AB as the hypotenuse. Now from the diagram, it is clear that the point of intersection of the other two legs of the right triangle having AB as the hypotenuse can be either P or Q.
CASE 1: When ∠ APB is taken.
The perpendicular sides in ∠ APB are AP and PB.
Now, side PB is parallel to x-axis and at a distance of 1 units above x-axis. So, equation of PB is, y=1 or y-1=0.
The side AP is parallel to y-axis and at a distance of 1 units on the right of y-axis. So, equation of AP is x=1 or x-1=0.
CASE 2: When ∠ AQB is taken.
The perpendicular sides in ∠ AQB are AQ and QB.
Now, side AQ is parallel to x-axis and at a distance of 3 units above x-axis. So, equation of AQ is, y=3 or y-3=0.
The side QB is parallel to y-axis and at a distance of 4 units on the left of y-axis. So, equation of QB is x=-4 or x+4=0.
Hence, the equation of the legs are :
x=1, y=1 or x=-4, y=3
Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.
The equation of the given line is
x + 3y = 7 (1)
Let point B (a, b) be the image of point A (3, 8). Accordingly, line (1) is the perpendicular bisector of AB.
The mid-point of line segment AB will also satisfy line (1). Hence, from equation (1), we have
On solving equations (2) and (3), we obtain a = β1 and b = β4.
Thus, the image of
the given point with respect to the given line is (β1,
β4).
If the lines y = 3x + 1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.
The equations of the given lines are
y = 3x + 1 (1) 2y = x + 3 (2)
y = mx + 4 (3)
Slope of line (1), m1 = 3
Slope of line (2), Slope of line (3), m3 = m
It is given that lines (1) and (2) are equally inclined to line (3). This means that the angle between lines (1) and (3) equals the angle between lines (2) and (3).
Thus, the
required value of m is .
If sum of the perpendicular distances of a variable point P (x, y) from the lines x + y - 5 = 0 and 3x - 2y + 7 = 0 is always 10. Show that P must move on a line.
The equations of the given lines are
x + y β 5 = 0 (1)
3x β 2y + 7 = 0 (2)
It is given that .
, which is the equation of a line.
Similarly, we can obtain the equation of line for any signs of . Thus, point P must move on a line.
Find equation of the line which is equidistant from parallel lines 9x + 6y - 7 = 0 and 3x + 2y + 6 = 0.
The equations of the given lines are 9x + 6y β 7 = 0 (1)
3x + 2y + 6 = 0 (2)
The perpendicular distance of P (h, k) from line (2) is given by
Since P (h, k) is equidistant from lines (1) and (2),
∴
9h + 6k β 7 = β 9h β 6k β 18
⇒ 18h + 12k + 11 = 0
Thus, the
required equation of the line
is 18x + 12y + 11 = 0.
A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.
Let the coordinates of point A be (a, 0). Draw a line (AL) perpendicular to the x-axis.
We know that angle of incidence is equal to angle of reflection. Hence, let
∠BAL = ∠CAL = Φ
Let ∠CAX = θ
∴∠OAB = 180° β (θ + 2Φ) = 180° β [θ + 2(90° β θ)]
= 180° β θ β 180° + 2θ
= θ
From equations (1) and (2), we obtain
Thus, the coordinates of point A are .
Prove that the product of the lengths of the perpendiculars drawn from the
On multiplying equations (2) and (3), we obtain
Hence, proved.
A person standing at the junction (crossing) of two straight paths represented by the equations 2x - 3y + 4 = 0 and 3x + 4y - 5 = 0 wants to reach the path whose equation is 6x - 7y + 8 = 0 in the least time. Find equation of the path that he should follow.
The equations of the given lines are 2x β 3y + 4 = 0 (1)
3x + 4y β 5 = 0 (2)
6x β 7y + 8 = 0 (3)
The person is standing at the junction of the paths represented by lines (1) and (2).
On solving equations (1) and (2), we obtain . Thus, the person is standing at point .
∴Slope of the line perpendicular to line (3)
Hence, the path that the person should follow is .