If the circles x^{2}+y^{2}+5Kx+2y+K=0 and 

2(x^{2}+y^{2})+2Kx+3y-1=0 , (K\epsilon R) , intersect

at the points P and Q , then the line 4x+5y-K=0 passes

through 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

 

Answers (1)

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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