If y² + py + 9 = 0 and y² + qy - 9 = 0 have common roots, then p² - q² is equal to?

Question

Question

Answer 1

Calculation:

We have, y² + py + 9 = 0 and y² + qy - 9 = 0

Let α be a common root of both equations

So α² + pα + 9 = 0 and ....(1)

α² + qα - 9 = 0 or α² = 9 - qα ....(2)

Putting (ii) in (i), we get (9 - qα) + pα + 9 = 0

α(p - q) = -18 or α = \(\rm \frac{18}{q - p}\)

Substituting in (i), \( \rm \frac{324}{(q - p)^{2}} + \frac{18p}{(q - p)} + 9 = 0\)

9(q - p)² + 18p(q - p) + 324 = 0

(q - p)2 + 2p(q - p) + 36 = 0

q² + p² - 2pq + 2pq - 2p² + 36 = 0

q² - p² + 36 = 0

p²- q² = 36