Let x = (2 ^ -1 ) and y = (1 2 ^ -1 4 3 ). If S = R : y ^ 2 =1- x… - Vidyadip

Question

Let $x =\sin \left(2 \tan ^{-1} \alpha\right)$ and $y =\sin \left(\frac{1}{2} \tan ^{-1} \frac{4}{3}\right)$. If $S =\left\{\alpha \in R : y ^{2}=1- x \right\}$, then $\sum_{\alpha \in S } 16 \alpha^{3}$ is equal to $...........$

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Answer

d

$\because \quad x=\sin \left(2 \tan ^{-1} \alpha\right)=\frac{2 \alpha}{1+\alpha^{2}}$

and $y=\sin \left(\frac{1}{2} \tan ^{-1} \frac{4}{3}\right)=\sin \left(\sin ^{-1} \frac{1}{\sqrt{5}}\right)=\frac{1}{\sqrt{5}}$

Now, $y^{2}=1-x$

$\frac{1}{5}=1-\frac{2 \alpha}{1+\alpha^{2}}$

$1+\alpha^{2}=5+5 \alpha^{2}-10 \alpha$

$2 \alpha^{2}-5 \alpha+2=0$

$\therefore \quad \alpha=2, \frac{1}{2}$

$\therefore \quad \sum_{\alpha \in S} 16 \alpha^{3}=16 \times 2^{3}+16 \times \frac{1}{2^{3}}$

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