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  4. The normal to the curve x=a(1+cosθ text), y=a sin θ at θ always passes through the fixed point

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Q. The normal to the curve $ x=a(1+cos\theta \text{)}, $ $ y=a\sin \theta $ at $ \theta $ always passes through the fixed point

JamiaJamia 2007

Solution:

We have, $ x=a(1+\cos \theta ),y=a\sin \theta $ $ \frac{dx}{d\theta }=a(-\sin \theta ),\frac{dy}{d\theta }=a\cos \theta $ $ \Rightarrow $ $ \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }=-\frac{\cos \theta }{\sin \theta } $ $ \therefore $ Equation of normal at $ [a(1+\cos \theta ).a\sin \theta ] $ $ (y-a\sin \theta )=\frac{\sin \theta }{\cos \theta }[x-a(1+\cos \theta )] $ It is clear that in the given options normal passes through the point (a, 0).