Question

If for some p, q, r ∈ R, not all have same sign, one of the roots of the equation (p2 + q2)x2 - 2q(p + r)x + q2 + r2 = 0 is also a root of the equation x2 + 2x - 8 = 0, then (q2 + r2)/p2 is equal to _______ .

Answer 1

Let α and β be the roots of (p2 + q2) x2 – 2q(p + r)x + q2 + r2 = 0
∴ α + β> 0 and αβ> 0
Also, it has a common root with x2 + 2x – 8 = 0
∴ The common root between above two equations is 4.
16(p2 + q2) – 8q(p + r) + q2 + r2 = 0
(16p2 – 8pq + q2) + (16q2 – 8qr + r2) = 0
(4p – q)2 + (4q – r)2 = 0
q = 4p and r = 16p
$∴\frac{q^2+r^2}{p^2}=\frac{16p^2+256p^2}{p^2}=272$
So, the correct is 272.