First slide

Tangents and normals

Question

The points on the curve $y^{2} = x^{3}$ where normal to the curve makes equal intercepts with the axes is /are

Moderate

A

$(4 / 9, \pm 8 / 27)$
B

$(3, \pm 3 \sqrt{3})$
C

$(1, \pm 1)$
D

None of these

Solution

Let foot of normal be $(x_{1}, y_{1})$ . Then $y_{1}^{2} = x_{1}^{3}$ …....(i)
Normal makes equal intercepts on axes

$\Rightarrow - \frac{d x}{d y} = \pm 1$ $\Rightarrow \frac{d x}{d y} = \pm 1$ $\Rightarrow {\frac{3 x^{2}}{2 y} |}_{(x_{1}, y_{1})} = \pm 1$

$\Rightarrow \frac{9 x_{1}^{4}}{4 y_{1}^{2}} = 1$ $\Rightarrow x_{1} = 4 / 9,$ $y_{1} = \pm 8 / 27$

Hence; required points are $(4 / 9, \pm 8 / 27)$ .

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