Let a b are roots of the equation x2-4x- 1-0 and c d are roots of the equation x2-36x- 2-0-The harmonic mean of the roots of the equation 7-2 x2- 7-5 x- 21-3-5 -0

The correct option is A 6 1α+1β =α+βα.β #160;where α and β are the roots of the given quadratic equation.Hence α+βα.β =7+ ...

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Byju's Answer

Standard XII

Mathematics

Sum of Infinite Terms of a GP

Let a b...

Question

Let $a$ & $b$ are roots of the equation $x^{2} - 4 x + α_{1} = 0$ and $c$ & $d$ are roots of the equation $x^{2} - 36 x + α_{2} = 0$ .
The harmonic mean of the roots of the equation $(7 + \sqrt{2}) x^{2} - (7 + \sqrt{5}) x + (21 + 3 \sqrt{5}) = 0$

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Solution

The correct option is A $6$
$\frac{1}{α} + \frac{1}{β}$ $= \frac{α + β}{α . β}$

where $α$ and $β$ are the roots of the given quadratic equation.

Hence
$\frac{α + β}{α . β}$
$= \frac{7 + \sqrt{5}}{21 + 3 \sqrt{5}} \times \frac{21 - 3 \sqrt{5}}{21 - 3 \sqrt{5}}$
$= \frac{1}{3}$ $= λ$

Hence
$H . M$
$= \frac{2}{λ}$
$= 2 \times 3$
$= 6$

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Let a & b are roots of the equation x2−4x+α1=0 and c & d are roots of the equation x2−36x+α2=0.The harmonic mean of the roots of the equation (7+√2)x2−(7+√5)x+(21+3√5)=0

The correct option is A 61α+1β=α+βα.β where α and β are the roots of the given quadratic equation.Henceα+βα.β=7+√521+3√5×21−3√521−3√5=13=λHenceH.M=2λ=2×3=6

Let $a$ & $b$ are roots of the equation $x^{2} - 4 x + α_{1} = 0$ and $c$ & $d$ are roots of the equation $x^{2} - 36 x + α_{2} = 0$ .
The harmonic mean of the roots of the equation $(7 + \sqrt{2}) x^{2} - (7 + \sqrt{5}) x + (21 + 3 \sqrt{5}) = 0$