If $x$ and $y$ are connected parametrically by the given equation, then without eliminating the parameter, find $\frac{d y}{d x}$ .
$x = a cos θ, y = b cos θ$

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Solution

The given equations are $x = a cos θ, y = b cos θ$
Then, $\frac{d x}{d θ} = \frac{d}{d θ} (a cos θ) = a (- sin θ) = - a sin θ$
and $\frac{d y}{d θ} = \frac{d}{d θ} (b cos θ) = b (- sin θ) = - b sin θ$
$∴ \frac{d y}{d x} = \frac{(\frac{d y}{d θ})}{(\frac{d x}{d θ})} = \frac{- b sin θ}{- a sin θ} = \frac{b}{a}$

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