Given x- a 2 - b 2 - a 2 - b 2 - a 2 - b 2 - a 2 - b 2 -Use componendo and dividendo to prove that - b 2 - 2 a 2 x x 2 -1-

x=√(a^2+b^2)+√(a^2 b^2)√(a^2+b^2) √(a^2 b^2) Use componendo and divivdendo rule: x+1x 1=√(a^2+b^2)+√(a^2 b^2)+√(a^2+b^2) √(a^2 b^2)√(a^2+b^2)+√(a^2 b^2 ...

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Byju's Answer

Standard X

Mathematics

Componendo and Dividendo

Given x=√ a...

Question

Given $x = \frac{\sqrt{a^{2} + b^{2}} + \sqrt{a^{2} - b^{2}}}{\sqrt{a^{2} + b^{2}} - \sqrt{a^{2} - b^{2}}}$ .
Use componendo and dividendo to prove that :
$b^{2} = \frac{2 a^{2} x}{x^{2} + 1}$ .

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Solution

$x = \frac{\sqrt{a^{2} + b^{2}} + \sqrt{a^{2} - b^{2}}}{\sqrt{a^{2} + b^{2}} - \sqrt{a^{2} - b^{2}}}$

Use componendo and divivdendo rule:

$\frac{x + 1}{x - 1} = \frac{\sqrt{a^{2} + b^{2}} + \sqrt{a^{2} - b^{2}} + \sqrt{a^{2} + b^{2}} - \sqrt{a^{2} - b^{2}}}{\sqrt{a^{2} + b^{2}} + \sqrt{a^{2} - b^{2}} - \sqrt{a^{2} + b^{2}} + \sqrt{a^{2} - b^{2}}}$

$\frac{x + 1}{x - 1} = \frac{2 \sqrt{a^{2} + b^{2}}}{2 \sqrt{a^{2} - b^{2}}}$

squaring on both sides

$\frac{(x + 1)^{2}}{(x - 1)^{2}} = \frac{2 (a^{2} + b^{2})}{2 (a^{2} - b^{2})}$

$\frac{(x + 1)^{2}}{(x - 1)^{2}} = \frac{(a^{2} + b^{2})}{(a^{2} - b^{2})}$

$(a^{2} - b^{2}) (x^{2} + 1 + 2 x) = (a^{2} + b^{2}) (x^{2} + 1 - 2 x)$

$4 a^{2} x - 2 b^{2} x^{2} - 2 b^{2} = 0$

$2 a^{2} x - b^{2} (x^{2} + 1) = 0$

$∴ b^{2} = \frac{2 a^{2} x}{x^{2} + 1}$

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Similar questions

Q. Given

x = \frac{\sqrt{a^{2} + b^{2}} + \sqrt{a^{2} - b^{2}}}{\sqrt{a^{2} + b^{2}} - \sqrt{a^{2} - b^{2}}}

use componendo and dividend0 to prove that :

b^{2} = \frac{2 a^{2} x}{x^{2} + 1}

Q. Given

x = \frac{\sqrt{a^{2} + b^{2}} + \sqrt{a^{2} - b^{2}}}{\sqrt{a^{2} + b^{2}} - \sqrt{a^{2} - b^{2}}}

, use componendo and dividendo to prove that

a^{2} - b^{2} = \frac{2 a^{2}}{x^{2} + 1}

Q. Given:

x = \frac{\sqrt{a^{2} + b^{2}} + \sqrt{a^{2} - b^{2}}}{\sqrt{a^{2} + b^{2}} - \sqrt{a^{2} - b^{2}}}

Use componendo and dividendo to prove that

b^{2} = \frac{{2 a x}^{2} x}{x^{3} + 1}

Q. The radius of the circle passing through the point of intersection of ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

and

x^{2} - y^{2} = 0

Q. If

sin α = \frac{a^{2} - b^{2}}{a^{2} + b^{2}}

then prove that

cot α = \frac{2 a b}{a^{2} - b^{2}}

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Given x=√a2+b2+√a2−b2√a2+b2−√a2−b2.Use componendo and dividendo to prove that :b2=2a2xx2+1.

Given $x = \frac{\sqrt{a^{2} + b^{2}} + \sqrt{a^{2} - b^{2}}}{\sqrt{a^{2} + b^{2}} - \sqrt{a^{2} - b^{2}}}$ .
Use componendo and dividendo to prove that :
$b^{2} = \frac{2 a^{2} x}{x^{2} + 1}$ .