Roots of equation (x^2 - bx) / (ax - c) = (p - 1) / (p + 1) are equal in magnitude but opposite in sign. Find p.

The equation given is

(x^2 - bx) / (ax - c) = (p - 1) / (p + 1)

=> (p + 1)(x^2 - bx) = (p - 1)(ax - c)

=> (p + 1)x^2 - bpx -bx = apx - ax - pc + c

=> (p + 1)x^2 + ax - bx - apx - bpx + pc - c = 0

=> (p + 1)x^2 + (a - b - ap - bp)x  + pc - c = 0

It is given that the roots are equal in magnitude but opposite in sign. Therefore, sum of them will be equal to 0. (for eg, t and -t are equal in magnitude but opposite in sign, and hence there sum will be equal to 0)

We know that Sum of roots of the equation ax^2 + bx + c = 0 is -b/a. So, here,

Sum of roots = -(a - b - ap - bp)/(p + 1)

=> 0 = -(a - b - ap - bp)/(p + 1)

=> 0 =  -(a - b - ap - bp)

=> 0 = (a - b - ap - bp)

=> ap + bp = a - b

=> p(a + b) = (a - b)

=> p = (a - b)/(a + b)

The value of p will be (a - b)/(a + b) if the roots are equal in magnitude but opposite in sign.