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Differentiation — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiation1 Mark
MCQ
If $\sin^{-1}\Big(\frac{\text{x}^2-\text{y}^2}{\text{x}^2+\text{y}^2}\Big)=\log\text{a}$ then $\frac{\text{dy}}{\text{dx}}$ is equal to:
  • A
    $\frac{\text{x}^2-\text{y}^2}{\text{x}^2+\text{y}^2}$
  • B
    $\frac{\text{y}}{\text{x}}$
  • ✓
    $\frac{\text{x}}{\text{y}}$
  • D
    $\text{None of these.}$
✓

Answer

Correct option: C.
$\frac{\text{x}}{\text{y}}$
$\sin^{-1}\Big(\frac{\text{x}^2-\text{y}^2}{\text{x}^2+\text{y}^2}\Big)=\log\text{a}$
$\frac{\text{x}^2-\text{y}^2}{\text{x}^2+\text{y}^2}=\sin(\log\text{a})=\text{k}$
$\text{x}^2-\text{y}^2=\text{k}(\text{x}^2+\text{y}^2)$
$\text{x}^2-\text{y}^2=\text{kx}^2+\text{ky}^2$
$\text{x}^2-\text{kx}^2=\text{ky}^2+\text{y}^2$
$(1-\text{k})\text{x}^2=(\text{x}+1)\text{y}^2$
$\frac{1-\text{x}}{\text{k}+1}=\frac{\text{y}^2}{\text{x}^2}\ .....(\text{i})$
Consider,
$\text{x}^2-\text{y}^2=\text{k}(\text{x}^2+\text{y}^2)$
Differentaiting with resepect to $x,$
$2\text{x}-2\text{y}\frac{\text{dy}}{\text{dx}}=2\text{kx}+2\text{ky}\frac{\text{dy}}{\text{dx}}$
$2\text{x}-2\text{kx}=2\text{ky}\frac{\text{dy}}{\text{dx}}+2\text{y}\frac{\text{dy}}{\text{dx}}$
$2\text{x}(1-\text{k})=2\text{y}(\text{k}+1)\frac{\text{dy}}{\text{dx}}$
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}(1-\text{k})}{\text{y}(\text{k}+1)}$
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}}{\text{y}}\times\frac{\text{y}^2}{\text{x}^2}\ .....(\because\text{from(i)})$
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}}{\text{y}}$

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