Question

If x sin theta and y = b tan theta then prove a²/x² -b² /y² = 1

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Answer 1

Solution :

• x = a sin ϑ
• y = b tan ϑ

LHS -

$:\implies\sf{ sin \theta = \dfrac{x}{a}}$

As we know ,

$:\implies\sf{ Cosec \theta = \dfrac{1}{sin \theta}}$

$:\implies\sf{ Cosec \theta = 1 \div \dfrac{x}{a}}$

$:\implies\sf{ Cosec \theta = \dfrac{a}{x}}$...(1)

RHS -

$:\implies\sf{ tan \theta = \dfrac{y}{a}}$

$:\implies\sf{ Cot \theta = \dfrac{1}{tan \theta}}$

$:\implies\sf{ Cot \theta = 1 \div \dfrac{y}{b}}$

$:\implies\sf{ Cot \theta = \dfrac{b}{y}}$...(2)

Now, As from our knowledge :

Cosec² ϑ - cot² ϑ = 1 ...(3)

Putting up the value of equation (1) and (2) in the equation (3) respectively.

$:\implies\sf{ \dfrac{y}{a} +\dfrac{y}{a} = 1}$

$:\implies\sf{\bigg(\dfrac{a}{x}\bigg)^{2} - \bigg( \dfrac{b}{y}\bigg)^{2} = 1}$

$:\implies\sf{\dfrac{a^{2}}{x^{2}} - \dfrac{b^{2}}{y^{2}} = 1}$

Hence , Proved !!