If the equation (a^2 + b^2)x^2 - 2(ac + bd)x + c^2 + d2 = 0 has e…

MCQ

If the equation $(a^2 + b^2)x^2 - 2(ac + bd)x + c^2 + d2 = 0$ has equal roots, then:

A
$\text{ab}=\text{cd}$
✓
$\text{ad}=\text{bc}$
C
$\text{ad}=\sqrt{\text{bc}}$
D
$\text{ab}=\sqrt{\text{cd}}$

✓

Answer

Correct option: B.

$\text{ad}=\text{bc}$

In the equation
$\Rightarrow (a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0$
$\Rightarrow D = B^2 - 4AC$
$\Rightarrow D = [-2(ac + bd)]^2 - 4(a^2 + b^2)(c^2 + d^2)$
$\Rightarrow D = 4[a^2c^2 + b2d^2 + 2abcd] - 4[a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2]$
$\Rightarrow D = 4a^2c^2 + 4b^2d^2 + 8abcd - 4a^2c^2 - 4a^2d^2 - 4b^2c^2 - 4b^2d^2$
$\Rightarrow D = 8abcd - 4a^2d2 - 4b^2c^2$
$\Rightarrow D = -4[a^2d^2 + b^2c^2 - 2abcd]$
$\Rightarrow D = -4(ad - bc)^2$
$\because$ Roots are equal
$\therefore D = 0$
$\Rightarrow - 4(ad - bc)^2 = 0$
$\Rightarrow ad - bc = 0$
$\Rightarrow ad = bc$

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