First slide

Tangents and normals

Question

$The tangent at the point (2, - 2) to the curve, x^{2} y^{2} - 2 x = 4 (1 - y) does not pass through$ the point

Moderate

A

$(4, \frac{1}{3})$
B

$(8, 5)$
C

$(- 4, - 9)$
D

$(- 2, - 7)$

Solution

$x^{2} y^{2} - 2 x = 4 - 4 y$
$D i f f e r e n t i a t e w . r . t . x$
$2 x y^{2} + 2 y . x^{2} . \frac{d y}{d x} - 2 = - 4. \frac{d y}{d x}$
$\Rightarrow \frac{d y}{d x} (2 y . x^{2} + 4) = 2 - 2 x . y^{2}$
$\Rightarrow {\frac{d y}{d x}|}_{(2, - 2)} = \frac{2 - 2 \times 2 \times 4}{2 (- 2) \times 4 + 4} = \frac{- 14}{- 12} = \frac{7}{6}$
$∴ E q u a t i o n o f \tan g e n t i s$
$(y + 2) = \frac{7}{6} (x - 2) \Rightarrow 7 x - 6 y = 26$
$(- 2, - 7) does not passes through the required tangent.$

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