Question

If the circles $x^2+y^2+2ax+cy+a=0$ and $x^2+y^2-3ax+dy-1=0$ interesect in two distinct points P and Q then the line 5x + by - a = 0 passes through P and Q for
(2005)

• A. exactly for 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

$S_1=x^2+y^2+2ax+cy+a=0$
$S_2=x^2+y^2-3ax+dy-1=0$
Equation of radical axis of $S_1$ and $S_2$
$S_1-S_2=0\Rightarrow 5ax+(c-d)y+a+1=0$
Given that $5x+by-a=0$ passes through P and Q

$\Rightarrow \frac{a}{1}=\frac{c-d}{b}=\frac{a+1}{-a}\Rightarrow a+1=-a^2$
$\Rightarrow a^2+a+1=0\Rightarrow a\notin R.$
Not real value of a.