Question

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 $Q$, then the line $5x+by-a=0$ passes through $P$ and $Q$ for

• A. Exactly two values of $a$
• B. Infinitely many values of $a$
• C. No value of $a$
• D. Exactly one value of $a$

Answer 1

Answer: (C)

No value of $a$

Given, equation of circles are

$S_1\equiv x^2+y^2+2ax+cy+a=0$

and $S_2\equiv x^2+y^2-3ax+dy-1=0$

chord through intersection points $P$ and $Q$ of the circles $S_1=0$ and $S_2=0$ is $S_1-S_2=0$

$\Rightarrow 5ax+(c-d)y+a+1=0$

On comparing 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 $a$.

Therefore,the line will passes through $P$ and $Q$ for no value of $a$.