Question

If the circles $x^2+y^2+5Kx+2y+K=0$ and $2(x^2+y^2)+2Kx+3y-1=0,(K\in R),$ intersect at the points $P$ and $Q$, then the line $4x+5y-K=0$ passes through $P$ and $Q$, for :

• A. exactly one value of $K$
• B. exactly two values of $K$
• C. infinitely many values of $K$
• D. no value of $K$

Answer 1

The correct option is D no value of $K$

Since, the two circles intersects each other at $P$ and $Q$ therefore, they lie on the common chord of the given circles.
Equation of the common chord : $S_1-S_2=0$
$\Rightarrow 4kx+\frac{y}{2}+K+\frac{1}{2}=0$
We are given the above equation as $4x+5y-K=0$
$\Rightarrow \frac{4k}{4}=\frac{\frac{1}{2}}{5}=\frac{k+\frac{1}{2}}{-k}$
$\Rightarrow$There is no value of $K$ for which satisfies the above condition.