If the roots of the equation (a 2 + b 2 )x 2 - 2(ab + cd)x + (c 2 + d 2 ) = 0 are equal, prove that .

Question

Philoid · Accepted Answer

For a quadratic equation, ax² + bx + c = 0,

D = b² – 4ac

If D = 0, roots are equal. For the given equation D would be -

(a² + b²) - 2(ab + cd) + (c² + d²) = 0

⇒ 4(ac + bd)² – 4(a² + b²)(c² + d²) = 0

⇒ a²c² + b²d² + 2acbd – a²d² – a²c² – b²d² – b²c² = 0

⇒ a²d² + b²c² – 2abcd = 0

⇒ (ad – bc) = 0

⇒ a/b = c/d