# If the circles $x^{2}+y^{2}+5Kx+2y+K=0$ and $2(x^{2}+y^{2})+2Kx+3y-1=0$ , $(K\epsilon R)$ , intersectat the points P and Q , then the line $4x+5y-K=0$ passesthrough P and Q , for : Option 1) infinitely many values of $K$ Option 2) no value of $K$ Option 3) exactly two values of $K$ Option 4) exactly one value of $K$

Given two circles are

$S_{1}=$ $x^{2}+y^{2}+5Kx+2y+K=0$

$S_{2}=$ $2(x^{2}+y^{2})+2Kx+3y-1=0$

Equation of common chord

$S_{1}-S_{2}=0$

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

Given equation of chord is

$4x+5y-k=0$..................................(2)

On Comparing (1) & (2)

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

There is no value of k

So, option (2) is correct.

Option 1)

infinitely many values of $K$

Option 2)

no value of $K$

Option 3)

exactly two values of $K$

Option 4)

exactly one value of $K$

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