State true or false.

The equation $2 a 2 + b 2 x 2 + 2 (a + b) x + 1 = 0$ has no real roots, when $a ≠ b$ .

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True

False

answer is A.

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Detailed Solution

Given,
The quadratic equation is

2 a 2 + b 2 x 2 + 2 (a + b) x + 1 = 0

.
Compare given equation with

a x 2 + b x + c = 0

a = 2 a 2 + b 2 b = 2 (a + b) c = 1

Let D be the discriminant of quadratic equation.
Question Image

⇒ D = − 4 a 2 − 4 b 2 + 8 a b ⇒ D = − 4 a 2 + b 2 − 2 a b ⇒ D = − 4 (a − b) 2

From the discriminant it is clear that

− 4

is a negative number and

(a − b) 2

is greater than zero, so the product of a negative and positive number is always a negative number it means the number is strictly less than zero.
As the discriminant is strictly less than zero, so the roots of given quadratic equation are not real.
Hence, it is true that the roots of the given quadratic equation are not real.
Therefore, the correct option is 1.

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