If the normals to the parabola `y^2=4a x` at three points `(a p^2,2a p),` and `(a q^2,2a q)` are concurrent, then the common root of equations `P x^2+q x+r=0` and `a(b-c)x^2+b(c-a)x+c(a-b)=0` is (a)`p` (b) `q` (c) `r` (d) `1`

Options

A. p
B. q
C. r
D. 1

Answer

D

Explanation

(4) Normal at points `(ap^(2),2ap),(aq^(2),2aq),and(ar^(2),2ar)` are concurrent.
Hence, the points are co-normal points. Therefore,
p+qr=0
So, `px^(2)+qx+r=0` has one root which is x=1.
Therefore, the common root is 1, which also satisfies the second equation.