If the equation has equal roots, where a + c = 15 and , then is equal to __________. [2025]

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Solution

Q.
If the equation $a (b - c) x^{2} + b (c - a) x + c (a - b) = 0$ has equal roots, where a + c = 15 and $b = \frac{36}{5}$ , then $a^{2} + c^{2}$ is equal to __________. [2025]

Ans.
(117)

Since, $a (b - c) x^{2} + b (c - a) x + c (a - b) = 0$ has equal roots

Put x = 1 in given equation.

$∴$ ab – ac + bc – ab + ac – bc = 0

$∴$ Both roots of the equation is 1.

$∴ 1 + 1 = \frac{- b (c - a)}{a (b - c)}$

$\Rightarrow 2 a b - 2 a c = - b c + a b$

$\Rightarrow 2 a c = b c + a b \Rightarrow 2 a c = b (a + c)$

$\Rightarrow 2 a c = \frac{36}{5} \times 15 = 108$

Also, a + c = 15 [Given]

$∴ a^{2} + c^{2} + 2 a c = 225$

$\Rightarrow a^{2} + c^{2} = 225 - 108 \Rightarrow a^{2} + c^{2} = 117$ .

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