First slide

Properties of ITF

Question

If $\cos^{- 1} \frac{x}{a} + \cos^{- 1} \frac{y}{b} = \frac{5 π}{12}$ and $\sin^{- 1} \frac{x}{a} - \sin^{- 1} \frac{y}{b} = \frac{π}{12}$ then $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} =$

Easy

A

1
B

$\frac{1}{4}$
C

$\frac{3}{4}$
D

$\frac{5}{4}$

Solution

$\cos^{- 1} \frac{x}{a} + \cos^{- 1} \frac{y}{b} = \frac{5 π}{12}$ ………(1)
$\cos^{- 1} \frac{y}{b} - \cos^{- 1} \frac{x}{a} = \frac{π}{12}$ ……..(2)
$\begin{array}{l} (1) + (2) \Rightarrow \cos^{- 1} \frac{y}{b} = \frac{π}{4} \Rightarrow \frac{y}{b} = \frac{1}{\sqrt{2}} \\ (1) - (2) \Rightarrow \cos^{- 1} \frac{x}{a} = \frac{π}{6} \Rightarrow \frac{x}{a} = \frac{\sqrt{3}}{2} \end{array}$
$∴ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = \frac{5}{4}$

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