• # question_answer If the circles ${{x}^{2}}+{{y}^{2}}+5Kx+2y+K=0$and $2\left( {{x}^{2}}+{{y}^{2}} \right)+2Kx+3y1=0,(K\in R),$ intersect at the points P and Q, then the line $4x+5yK=0$ passes through P and Q for : [JEE Main 10-4-2019 Morning] A) exactly two values of KB) exactly one value of KC) no value of K.D) infinitely many values of K

Equation of common chord $4kx+\frac{1}{2}y+k+\frac{1}{2}=0$                              ....(1) and given line is $4x+5yk=0$          .....(2) On comparing (1) & (2), we get $k=\frac{1}{10}=\frac{k+\frac{1}{2}}{-k}$$\Rightarrow$No real value of k exist