Q104 / 160 · AP EAMCET 2024 (21 May Shift 2) · Mathematics

If $y=\sin ^{-1} x$ then $\left(1-x^2\right) y_2-x y_1=$

  • Correct answer
    A
    0
  • B
    1
  • C
    2
  • D
    $2 y$

Solution

$\begin{aligned} & \text { } y=\sin ^{-1} x \Rightarrow x=\sin y \Rightarrow \cos y \frac{d y}{d x}=1 \\ & \Rightarrow \frac{d y}{d x}=\sec y \Rightarrow \frac{d^2 y}{d x^2}=\sec y \tan y \frac{d y}{d x}=\frac{\sin y}{\cos ^2 y} \frac{d y}{d x} \\ & \Rightarrow\left(1-\sin ^2 y\right) y_2=\sin y \frac{d y}{d x} \Rightarrow\left(1-x^2\right) y_2=x y_1 \\ & \Rightarrow\left(1-x^2\right) y_2-x y_1=0 .\end{aligned}$

Hence, the correct answer is option: 'A'.

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