Question

lf 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 one value of $a$
• B. no value of $a$
• C. infinitely many values of $a$
• D. exactly two values of $a$

Answer 1

The correct option is B no value of $a$

$S_1=x^2+y^2+2ax+cy+a=0$
$S_2=x^2+y^2-3ax+dy-1=$
Equation of PQ will be radical axis of which is given by, $S_1-S_2=0$
$\Rightarrow 5ax+(c-d)y+(a+1)=0$
Given that $5x+by-a=0$ passes
through $P$ and $Q$
$\Rightarrow \frac{a}{1}=\frac{c-d}{b}=\frac{a+1}{-a}\Rightarrow a+1=-a^2$
$\Rightarrow a^2+a+1=0\Rightarrow$ No real value of $a$.