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State whether the following statement is true or false.

The roots of the equation $a x^{2} + bx + c = 0$ will be reciprocal of each other if $c \neq a$ .

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True

False

answer is B.

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Detailed Solution

The standard form of a quadratic equation is

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

where

a, b c

are real numbers and

a \neq 0

.
And,
The sum of the roots is given by

- \frac{b}{a}

.
Product of the roots is given by

\frac{c}{a}

.
As we know, a quadratic equation has two roots. Let’s consider one of the roots of the equation is

α

and according to the given condition, the second root of the equation is the reciprocal of the first one so, the second root will be

\frac{1}{α}

.
Then,
Product of the roots =

\frac{c}{a}

⟹ α \times \frac{1}{α} = \frac{c}{a}

⟹ 1 = \frac{c}{a}

∴ a = c

i . e . c = a

∴

The roots of the equation

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

will be reciprocal of each other if c = a.
Hence, the given statement is false.
Therefore, option (2) is correct.

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