First slide

Tangents and normals

Question

$The equations of the tangents at the origin to the curve y^{2} = x^{2} (1 + x + x^{2}) is$

Moderate

A

$x = \pm y$
B

$2 x = \pm y$
C

$3 x = \pm y$
D

$2 y = \pm x$

Solution

$Given curve is y^{2} = x^{2} (1 + x + x^{2})$
$\Rightarrow y = \pm \sqrt{x^{2} + x^{3} + x^{4}} \Rightarrow \frac{d y}{d x} = \pm \frac{2 x + 3 x^{2} + 4 x^{3}}{2 \sqrt{x^{2} + x^{3} + x^{4}}} = \pm \frac{2 + 3 x + 4 x^{2}}{2 \sqrt{1 + x + x^{2}}} a t (0, 0), m = \pm 1 e q u a t i o n o f \tan g e n t a t (0, 0) i s y = \pm x$

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