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

$x = a c o s θ, y = b c o s θ$

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Solution

Given, $x = a c o s θ, y = b c o s θ$

Differentiating w.r.t. $θ, w e g e t \frac{d x}{d θ} = a (- s i n θ) a n d \frac{d y}{d θ} = b (- s i n θ)$

$∴ \frac{d y}{d x} = \frac{\frac{d y}{d θ}}{\frac{d x}{d θ}} = \frac{d y}{d θ} \times \frac{d θ}{d x} = \frac{- b s i n θ}{- a s i n θ} = \frac{b}{a} (∵ \frac{d y}{d x} = \frac{d y / d t}{d x / d t})$

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