Q5 of 227 Page 110

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

Given: (a2 + b2) x2 – 2 (ac + bd) x + c2 + d2 = 0


To prove:


Proof:


We know that,


D = b2 – 4ac


If roots are equal, then b2 = 4ac


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


⇒ 4(a2c2 + b2d2 + 2acbd) = 4 (a2c2 + a2d2 + b2c2 + b2d2)


⇒ 2acbd = a2d2 + b2c2


⇒ a2d2 + b2c2 – 2acbd = 0


⇒ (ad – bc)2 = 0


⇒ ad – bc = 0


⇒ ad = bc


⇒


Hence proved.


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