If (a^2 + b^2)x^2 + 2(ab + bd)x + c^2 + d^2 = 0 has no real roots…

MCQ

If $(a^2 + b^2)x^2 + 2(ab + bd)x + c^2 + d^2 = 0$ has no real roots, then:

A
$ab = bc$
B
$ab = cd$
C
$ac = bd$
✓
$\text{ad}\neq\text{bc}$

✓

Answer

Correct option: D.

$\text{ad}\neq\text{bc}$

The given quadric equation is $(a^2 + b^2)x^2 + 2(ab + bd)x + c^2 + d^2 = 0$, and roots are equal.
Here, $a = (a^2 + b^2), b = 2(ab + bd)$ and, $c = c^2 + d^2$
As we know that $D = b^2 - 4ac$
Putting the value of $a = (a^2 + b^2), b = 2(ab + bd)$ and, $c = c^2 + d^2$
$= {2(ab + bd)}^2 - 4 \times (a^2 + b^2) \times (c^2 + d^2)$
$= 4a^2b^2 + 4b^2d^2 + 8ab^2d - 4(a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2)$
$= 4a^2b^2 + 4b^2d^2 + 8ab^2d - 4a^2c^2 - 4a^2d^2 - 4b^2c^2 - 4a^2d^2$
$= 4a^2b^2 + 8ab^2d - 4a^2c^2 - 4a^2d^2 - 4b^2c^2$
$= 4(a^2b^2 + 2ab^2d - a^2c^2 - a^2d^2 - b^2c^2)$
The given equation will have no real roots, if $D < 0$
$4(a^2b^2 + 2ab^2d - a^2c^2 - a^2d^2 - b^2c^2) < 0$
$a^2b^2 + 2ab^2d - a^2c^2 - a^2d^2 - b^2c^2 < 0$
$\text{ad}\neq\text{bc}$
Thus, the correct answer is $(d)$

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