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)

If $S_1=0$ and $S_2=0$ are two circles, then the equation of line passes through the points of intersection of $S_1=0$ and $S_2=0$, is $S_1-S_2=0$.
Equation of circles are
$S_1=x^2+y^2+2ax+cy+a=0$
and $S_2=x^2+y^2-3ax+dy-1=0$
respectively.
Chord through intersection points P and Q of the circles $S_1=0$ and $S_2=0$ is $S_1-S_2=0$.
$\therefore 5ax+\left(c-d\right)y+a+1=0$
On comparing it with $5x+by-a=0$, we get
$\frac{5a}{5}=\frac{c-d}{b}=\frac{a+1}{-a}$
$\Rightarrow a\left(-a\right)=a+1\Rightarrow a^2+a+1=0$
Which gives no real value of a.
$\therefore$ The line will passes through P and Q for no value of a