If the solve the problem

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

exactly two values of K
exactly one value of K
no value of K.
infinitely many values of K

Correct Option: , 3

Solution:

Equation of common chord

$4 \mathrm{kx}+\frac{1}{2} \mathrm{y}+\mathrm{k}+\frac{1}{2}=0$ .....(1)

and given line is $4 x+5 y-k=0$ .......(2)

On comparing (1) & (2), we get

$k=\frac{1}{10}=\frac{k+\frac{1}{2}}{-k}$

$\Rightarrow$ No real value of $k$ exist

If the solve the problem

Leave a comment

All Study Material