If the circles $x^{2} + y^{2} + 2ax + cy + a = 0$ and $x^{2} + y^{2} - 3ax + dy - 1 = 0$ intersect in two distinct points $P$ and $\mathcal{Q}$, then the line $5x + by - a = 0$ passes through $P$ and $\mathcal{Q}$ for

Question

Dr. Abubacker · Accepted Answer

(c)
Chord through intersection points $P$ and $\mathcal{Q}$ of the given circles is $S_{1} - S_{2} = 0$
$	herefore\ \ \left( x^{2} + y^{2} + 2ax + cy + a ight) - \left( x^{2} + y^{2} - 3ax + dy - 1 ight) = 0$
$\Rightarrow \ 5ax + (c - d)y + a + 1 = 0$
On comparing it with $5x + by - a = 0$, we get
$\frac{5a}{5} = \frac{c - d}{b} = \frac{a + 1}{- a}$
$\Rightarrow \ \ \ a( - a) = a + 1$
$\Rightarrow \ \ a^{2} + a + 1 = 0$
Which gives no real value of $a$
Hence, the line will passes through $P$ and $\mathcal{Q}$ for no value of $a$

Q. If the circles $x^{2} + y^{2} + 2ax + cy + a = 0$ and $x^{2} + y^{2} - 3ax + dy - 1 = 0$ intersect in two distinct points $P$ and $\mathcal{Q}$, then the line $5x + by - a = 0$ passes through $P$ and $\mathcal{Q}$ for

Explanation

Reviewed by Dr. Abubacker

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