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Roots of a Quadratic Equation

Let a, b, c...

Question

Let $a, b, c, d$ be distinct real numbers and $a$ and $b$ are the roots of quadratic equation $x^{2} - 2 c x - 5 d = 0$ . If $c$ and $d$ are the roots of the quadratic equation $x^{2} - 2 a x - 5 b = 0$ .then find the numerical values of $a + b + c + d$ .

30

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15

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10

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60

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Solution

The correct option is A $30$
$a, b$ are roots of $x^{2} - 2 c x - 5 d = 0$
$\Rightarrow a + b = 2 c$
and $c, d$ are roots of $x^{2} - 2 a x - 5 b = 0$
$\Rightarrow c + d = 2 a$
$∴ (a - c) + (b - d) = 2 (c - a)$
$\Rightarrow (b - d) = 3 (c - a) \dots (1)$

Since $c$ is the root of $x^{2} - 2 a x - 5 b = 0 \Rightarrow c^{2} - 2 a c - 5 b = 0 \dots (2)$

Similarly $a$ is root of $x^{2} - 2 c x - 5 d = 0 \Rightarrow a^{2} - 2 a c - 5 d = 0 \dots (3)$

On subtracting (3) from (2)
$c^{2} - a^{2} = 5 (b - d)$

$\Rightarrow (c - a) (c + a) = 5 \times 3 (c - a) \dots \dots (f r o m (1))$
$\Rightarrow c + a = 15$

$∴ a + b + c + d = 2 c + 2 a$

$= 2 (c + a) = 2 \times 15 = 30$

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Let a,b,c,d be distinct real numbers and a and b are the roots of quadratic equation x2−2cx−5d=0. If c and d are the roots of the quadratic equation x2−2ax−5b=0 .then find the numerical values of a+b+c+d.

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