If x = a , y = b then d y d x = ? - Vidyadip

MCQ

If $x = a \sec \theta, y = b \tan \theta$ then $\frac{d y}{d x}=$ ?

A
$\frac{b}{a} \sec \theta$
B
$\frac{b}{a} \tan \theta$
✓
$\frac{b}{a} \operatorname{cosec} \theta$
D
$\frac{b}{a} \cot \theta$

✓

Answer

Correct option: C.

$\frac{b}{a} \operatorname{cosec} \theta$

$x = a \sec \theta$, we get
$\therefore \frac{d x}{d \theta}=a \sec \theta \cdot \tan \theta$
$\therefore \frac{d \theta}{d x}=\frac{1}{asec \theta \cdot \tan \theta}$
$y=b \tan \theta,$ we get
$\therefore \frac{d y}{d \theta}=b \cdot \sec ^2 \theta$
$\Rightarrow \frac{d y}{d x}=\frac{d y}{d \theta} \times \frac{d \theta}{d x}$
$\Rightarrow \frac{d y}{d x}=b \cdot \sec ^2 \theta \times \frac{1}{asec \theta \cdot \tan \theta}$
$\Rightarrow \frac{d y}{d x}=\frac{b \sec \theta}{a \tan \theta}$
$\Rightarrow \frac{dy}{dx}=\frac{b \cdot \frac{1}{\cos \theta}}{a \cdot \frac{\sin \theta}{\cos \theta}}$
$\Rightarrow \frac{d y}{d x}=\frac{b}{a} \operatorname{cosec} \theta$

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