Question

If the roots of the equation (a2 + b2) x2 – 2(ac + bd) x + (c2 + d2) = 0 are equal, prove that /
= /
.

Answer 1

Full question :

If the roots of the equation (a² + b²) x² – 2(ac + bd) x + (c² + d²) = 0 are equal, prove that a/b = c/d

.

Solution :

( a² + b² ) x² - 2 ( ac + bd ) x + ( c² + d² ) = 0

Comparing with ax² + bx + c = 0 we get :

a = a² + b²

b = - 2 ( ac + bd )

c = c² + d²

When roots are equal :

b² = 4 ac

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

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

Use the expansion of ( a + b )² = a² + b² + 2 ab

⇒ 4 ( ac + bd )² = 4 ( a²c² + a²d² + b²c² + b²d² )

Cancel 4 both sides :

⇒  ( a²c² + b²d² + 2 abcd ) = a²c² + a²d² + b²d² + b²c²

⇒ a²d² + b²c² - 2 abcd = 0

Use the expansion of a² + b² - 2 ab = ( a - b )²

⇒ ( ad - bc )² = 0

Take square root both sides :

⇒ ad - bc = 0

⇒ ad = bc

⇒ a/b = c/d

Hence proved !

Answer 2

Your question is incomplete

Complete question

If the roots of the equation (a2 + b2) x2 – 2(ac + bd) x + (c2 + d2) = 0 are equal, prove that

a/b= c/d.

refer to attachment for answer

thanks