If - - are the roots of the equation a2-b2 x2-2 xa c-b d-c2-d2-0- then - is equal to

The correct option is A 1(a^2+b^2)x^2+2(ac+bd)x+c^2+d^2 = 0 Roots are real. ⇒Δ≥ 0 nbsp; ⇒ 4(ac+bd)^2 4(a^2+b^2)(c^2+d^2)≥ 0 ⇒ (ac+bd)^2 [(ac)^2+(bd)^2+(bc)^ ...

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Byju's Answer

Standard XII

Mathematics

Harmonic Mean

If α, β∈ℝ are...

Question

If $α, β \in R$ are the roots of the equation $(a^{2} + b^{2}) x^{2} + 2 x (a c + b d) + c^{2} + d^{2} = 0$ , then $\frac{α}{β}$ is equal to

1

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2

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\frac{1}{2}

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\frac{1}{4}

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Solution

The correct option is A $1$
$(a^{2} + b^{2}) x^{2} + 2 (a c + b d) x + c^{2} + d^{2} = 0$
Roots are real.
$\Rightarrow Δ \geq 0$
$\Rightarrow 4 (a c + b d)^{2} - 4 (a^{2} + b^{2}) (c^{2} + d^{2}) \geq 0$
$\Rightarrow (a c + b d)^{2} - [(a c)^{2} + (b d)^{2} + (b c)^{2} + (a d)^{2}] \geq 0$
$\Rightarrow 2 a c b d - (b c)^{2} - (a d)^{2} \geq 0$
$\Rightarrow - (b c - a d)^{2} \geq 0$
$\Rightarrow (b c - a d)^{2} \leq 0$
$\Rightarrow (b c - a d)^{2} = 0$
$\Rightarrow b c = a d$
So, the roots are real and equal.

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If α,β∈R are the roots of the equation (a2+b2)x2+2x(ac+bd)+c2+d2=0, then αβ is equal to

The correct option is A 1(a2+b2)x2+2(ac+bd)x+c2+d2=0 Roots are real. ⇒Δ≥0 ⇒4(ac+bd)2−4(a2+b2)(c2+d2)≥0 ⇒(ac+bd)2−[(ac)2+(bd)2+(bc)2+(ad)2]≥0 ⇒2acbd−(bc)2−(ad)2≥0 ⇒−(bc−ad)2≥0 ⇒(bc−ad)2≤0 ⇒(bc−ad)2=0 ⇒bc=ad So, the roots are real and equal.

If $α, β \in R$ are the roots of the equation $(a^{2} + b^{2}) x^{2} + 2 x (a c + b d) + c^{2} + d^{2} = 0$ , then $\frac{α}{β}$ is equal to