First slide

Tangents and normals

Question

The slope of the normal to the curve
$x = a (\cos θ + θ \sin θ)$

$y = a (\sin θ - θ \cos θ)$ at any point $θ$ is

Moderate

A

$\tan θ$
B

$\cot θ$
C

$- \tan θ$
D

$- \cot θ$

Solution

$\frac{d x}{d θ} = a (- \sin θ + θ \cos θ + \sin θ)$
and $\frac{d y}{d θ} = a (\cos θ + θ \sin θ - \cos θ)$

Hence $\frac{d y}{d x} = (\frac{d y}{d θ}) / (\frac{d x}{d θ}) = \frac{a θ \sin θ}{a θ \cos θ} = \tan θ$ .

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