First slide

Properties of ITF

Question

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

Moderate

A

36
B

$- 36 \sin^{2} θ$
C

$36 \sin^{2} θ$
D

$36 \cos^{2} θ$

Solution

We have,
$\begin{array}{l} \cos^{- 1} \frac{x}{2} + \cos^{- 1} \frac{y}{3} = θ \\ \Rightarrow \cos^{- 1} \{\frac{x y}{6} - \sqrt{1 - \frac{x^{2}}{4}} \sqrt{1 - \frac{y^{2}}{9}}\} = θ \\ \Rightarrow x y - \sqrt{4 - x^{2}} \sqrt{9 - y^{2}} = 6 \cos θ \\ \Rightarrow (x y - 6 \cos θ)^{2} = (4 - x^{2}) (9 - y^{2}) \\ \Rightarrow - 12 x y \cos θ + 36 \cos^{2} θ = 36 - 4 y^{2} - 9 x^{2} \\ \Rightarrow 9 x^{2} + 4 y^{2} - 12 x y \cos θ = 36 \sin^{2} θ \end{array}$

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