If x = a c o s e c θ and y = b c o t θ , then which of the following equations is true ? Options: A . &nbsp x 2 − y 2 = a 2 − b 2 B . &nbsp x 2 a 2 − y 2 b 2 = 1 C . &nbsp x 2 + y 2 = a 2 + b 2 D . &nbsp x 2 a 2 + y 2 b 2 = 1

If x=a cosec θ and y=b cot θ, then which of th

Question

If $x = a c o s e c θ$ and $y = b c o t θ$ , then which of the following equations is true ?

Options:

A .

x^{2} - y^{2} = a^{2} - b^{2}

B .

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

C .

x^{2} + y^{2} = a^{2} + b^{2}

D .

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

Answer: Option B
:
B

$x = a c o s e c θ \Rightarrow \frac{x}{a} = c o s e c θ$ .....(1)

$y = b c o t θ \Rightarrow \frac{y}{b} = c o t θ$ .....(2)

Squaring the equations and subtracting (2) from (1) we get,

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = c o s e c^{2} θ - c o t^{2} θ$

$∴ \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1.... (1 + c o t^{2} θ = c o s e c^{2} θ)$

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