First slide

Properties of ITF

Question

If $\cos^{- 1} \frac{x}{2} + \cos^{- 1} \frac{y}{3} = θ$ , then $9 x^{2} - 12 xycos θ + 4 y^{2} =$

Moderate

A

36
B

– 36 sin² θ
C

36 sin² θ
D

36 cos² θ

Solution

Given, $θ = \cos^{- 1} (\frac{x}{2} \cdot \frac{y}{3} - \sqrt{1 - \frac{x^{2}}{4}} \sqrt{1 - \frac{y^{2}}{9}})$
$\begin{array}{l} \Rightarrow \cos θ = \frac{xy}{6} - \frac{\sqrt{4 - x^{2}} \sqrt{9 - y^{2}}}{6} \\ \Rightarrow (xy - 6 \cos θ)^{2} = (4 - x^{2}) (9 - y^{2}) \\ = 36 - 9 x^{3} - 4 y^{7} + x^{7} y^{7} \\ \Rightarrow 36 \cos^{2} θ - 12 xycos θ + 4 y^{2} + 9 x^{2} = 36 \\ ∴ 9 x^{2} - 12 xycos θ + 4 y^{2} = 36 \sin^{2} θ \end{array}$

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