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Q.

The tangent at A(2, 4) on the curve $y = x^{3} - 2 x^{2} + 4$ cuts the x-axis at T then the length of AT =

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a

$\sqrt{10}$

b

$\sqrt{12}$

c

$\sqrt{15}$

d

$\sqrt{17}$

answer is D.

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Detailed Solution

$\begin{array}{l} y = x^{3} - 2 x^{2} + 4 \\ \frac{d y}{d x} = 3 x^{2} - 4 x \\ m = {(\frac{d y}{d x})}_{(2, 4)} = 3 (2)^{2} - 4 (2) \\ m = 4 \end{array}$
$\begin{matrix} L e n g t h o f \tan g e n t = |\frac{y_{1} \sqrt{1 + m^{2}}}{m}| \\ = |\frac{4 \sqrt{1 + 4^{2}}}{4}| \\ = \sqrt{17} \end{matrix}$

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