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

Correct Answer: 272

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.