If the roots equal of the equation ax2-bx-c-0 are of the form aa-1 and a-1a then the va-b-c2 is

The correct option is C b^2 4ac products of roots = (a/a 1) ×(a+1/a)= c/a #160; (1).... 2a2 = c+ac a (by componendo amp; dividend o) (2 ...

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Byju's Answer

Standard XII

Mathematics

Sum of n Terms

If the roots ...

Question

If the roots equal of the equation $a x^{2} + b x + c = 0$ are of the form $\frac{a}{a - 1}$ and $\frac{a + 1}{a}$ then the $v (a + b + c)^{2}$ is

2 b^{2} - a c

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b^{2} - a c

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b^{2} - 4 a c

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4 b^{2} 2 a c

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Solution

The correct option is C $b^{2} - 4 a c$
products of roots $= (\frac{a}{a - 1}) \times (\frac{a + 1}{a}) = \frac{c}{a}$

$(1) . . . . \frac{2 a}{2} = \frac{c + a}{c - a}$ (by componendo & dividend o)

$(2) . . . . \frac{a + 1 - a + 1}{a - 1} = \frac{c - a}{a} \Rightarrow \frac{2}{a - 1} = \frac{c - a}{a}$ (by componendo dividendo)

$(1) \times (2) \Rightarrow \frac{a}{a - 1} = \frac{c + a}{2 a}$

Now this is a root of given equation.

$a x^{2} + 6 x + c = 0$

$a \frac{(c + a)^{2}}{4 a^{2}} + b \times [\frac{c + a}{2 a}] + c = 0$

$b^{2} + (c + a)^{2} + 2 b (c + a) + 4 a c = b^{2}$ (Add $b^{2}$ on both sides.)
$(a + b + c)^{2} = b^{2} - 4 a c .$

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Similar questions

Q. If roots of the equation

{a x}^{2} + b x + c = 0 a r e \frac{a}{a - 1} a n d \frac{a + 1}{a}

then prove that

(a + b + c)^{2} - b^{2} - 4 a c

Q. If roots of the equation

a x^{2} + b x + c = 0 a r e . \frac{α}{α - 1} . a n d . \frac{α + 1}{α}

, then prove that

(a + b + c)^{2} = b^{2} - 4 a c

The roots of a quadratic equation $a x^{2} + b x + c = 0$ are given by $\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , provided $b^{2} - 4 a c \geq 0$ .

Q. If the roots of

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

are

α, β

and the roots of

A X^{2} + B x + C = 0

are

(α - k), (β - k)

. Then

(\frac{B^{2} - 4 A C}{b^{2} - 4 a c})

is equal to

If b² – 4ac = 0 then The roots of the Quadratic equation ax² + bx + c = 0 are given by :

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If the roots equal of the equation ax2+bx+c=0 are of the form aa−1 and a+1a then the v(a+b+c)2 is

If the roots equal of the equation $a x^{2} + b x + c = 0$ are of the form $\frac{a}{a - 1}$ and $\frac{a + 1}{a}$ then the $v (a + b + c)^{2}$ is