First slide

Tangents and normals

Question

The tangent at any point on the curve x = a_{$\cos^{3} θ,$} y=a_{$\sin^{3} θ$} meets the axes in P and Q. The locus of the mid point of PQ is

Moderate

A

_{$x^{3 / 2} + y^{3 / 2} = a^{3 / 2}$}
B

_{$x^{2 / 3} + y^{2 / 3} = a^{2 / 3}$}
C

4(x+y) = a
D

_{$4 (x^{2} + y^{2}) = a^{2}$}

Solution

We have, _{$\frac{d y}{d x}$} =_{$\frac{\frac{d y}{d θ}}{\frac{d x}{d θ}} =$} _{$\frac{3 a \sin^{2} θ \cos θ}{3 a \cos^{2} θ (- \sin θ)} = - \tan θ$}
Equation of tangent at _{$' θ'$} is _{$y - a \sin^{3} θ$} =-tan_$θ$ (_{$x - a \cos^{3} θ$} )

_{$\Rightarrow$} _{$\frac{x}{a \cos θ} + \frac{y}{a \sin θ} = 1$}

_$∴$ P is _{$(a \cos θ, 0) a n d (0, a \sin θ) .$} If mid point of PQ is (h, k), then

_{$h = \frac{a \cos θ}{2}, k = \frac{a \sin θ}{2}$} Eliminating _$θ$ , we get

_{$h^{2} + k^{2} = \frac{a^{2}}{4}$}

_$∴$ focus is _{$x^{2} + y^{2} = \frac{a^{2}}{4}$}

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