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If the roots of the equation $a 2 b 2 x 2 − 2 b (a + c) x + b 2 + c 2 = 0$ are equal, then

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2b = a + c

2

= acd

b = 2 ac a + c

ad = bc

answer is D.

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Detailed Solution

Given:

a 2 + b 2 x 2 − 2 (a c + b d) x + c 2 + d 2 = 0

We know for real and equal roots; discriminant equals to zero.

b 2 − 4 a c = 0

.........

(1)

The given equation:

a 2 + b 2 x 2 − 2 (a c + b d) x + c 2 + d 2 = 0

.............

(2)

Compare equation (2) with a general quadratic equation

a x^{2} + bx + c = 0

a = a 2 + b 2, b = − 2 (a c + b d), c = c 2 + d 2

Substitute the values in equation (1)

⇒ (− 2 (a c + b d)) 2 − 4 a 2 + b 2 c 2 + d 2 = 0 ⇒ 4 (a c + b d) 2 − 4 a 2 c 2 + a 2 d 2 + b 2 c 2 + b 2 d 2 = 0

⇒ 4 (a c + b d) 2 − a 2 c 2 + a 2 d 2 + b 2 c 2 + b 2 d 2 = 0 ⇒ (a c + b d) 2 − a 2 c 2 + a 2 d 2 + b 2 c 2 + b 2 d 2 = 0

⇒ a 2 c 2 + b 2 d 2 + 2 a b c d − a 2 c 2 − a 2 d 2 − b 2 c 2 − b 2 d 2 = 0 ⇒ 2 a b c d − a 2 d 2 − b 2 c 2 = 0

⇒ − 1 − 2 a b c d + a 2 d 2 + b 2 c 2 = 0 ⇒ a 2 d 2 + b 2 c 2 − 2 a b c d = 0

⇒ (a d − b c) 2 = 0 ⇒ a d − b c = 0 ⇒ a d = b c

Therefore, if the roots of the equation

a 2 b 2 x 2 − 2 b (a + c) x + b 2 + c 2 = 0

are equal, then

ad = bc

.
Hence, the correct option is 4.

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