If the circles `x^2+y^2+2a x+c y+a=0` and points `Pa n dQ` , then find the values of `a` for which the line `5x+b y-a=0` passes through `Pa n dQdot`

Options

A. exactly two values of a
B. infinitely many values of a
C. no value of a
D. exactly one value of a

Answer

C

Explanation

The equation of the common chord of the given circles is
`5ax+(c-d)y+(a+1)=0 ...(i)` [Using: `S_(1)-S_(2)=0`]
But, the equation of the common chord is given as
5x+by-a=0 ...(ii)
Comparing (i) and (ii), we get
`(5a)/(5)=(c-d)/(d)=(a+1)/(-a)`
`rArr (a)/(1)=(c-d)/(b)=-(a+1)/(a) rArr a= - ((a+1))/(a) rArr a^(2)+a + 1 = 0`
Clearly, `a^(2)+a+1=0` has no real roots. So, there is no real value of a for which 5x+by-a=0 passes through P and Q.