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If the solve the problem


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 :

  1. exactly two values of K

  2. exactly one value of K

  3. no value of K.

  4. infinitely many values of K

Correct Option: , 3


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


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

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