Q. The equation of the parabola whose focus is $(3, - 4)$ and directrix $6x - 7y + 5 = 0$, is
A) $(7x + 6y)^{2} - 570x + 750y + 2100 = 0$
B) $(7x + 6y)^{2} + 570x - 750y + 2100 = 0$
C) $(7x - 6y)^{2} - 570x + 750y + 2100 = 0$
D) $(7x - 6y)^{2} + 570x - 750y + 2100 = 0$
Explanation
(a)
If $P(x,y)$ b ea point on a parabola, then by the definition of parabola
$(PS)^{2} = (PM)^{2}$
$\Rightarrow (x - 3)^{2} + (y + 4)^{2} = \left( \frac{6x - 7y + 5}{\sqrt{6^{2} + 7^{2}}} \right)^{2}$
$\Rightarrow \ \ 85\left( x^{2} - 6x + 9 + y^{2} + 8y + 16 \right)$
$\Rightarrow \ 36x^{2} + 49y^{2} + 25 - 84xy - 70y + 60x\$
$\Rightarrow \ (7x + 6y)^{2} - 570x + 750y + 2100 = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.