Find $\frac{d y}{d x}$ , if x and y are connected parametrically by the equations given in questions without eliminating the parameter.

$x = a s e c θ, y = b t a n θ .$

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Solution

Given, $x = a s e c θ, y = b t a n θ .$

Differentiating w.r.t.. $θ$ , we get

$\frac{d x}{d θ} = a s e c θ t a n θ a n d \frac{d y}{d θ} = b s e c^{2} θ ∴ \frac{d y}{d x} = \frac{\frac{d y}{d θ}}{\frac{d x}{d θ}} = \frac{d y}{d θ} \times \frac{d θ}{d x} = \frac{b s e c^{2} θ}{a s e c θ t a n θ} = \frac{b}{a} (\frac{s e c θ}{t a n θ}) = \frac{b}{a} c o s e c θ$

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