Q83 / 125 · BITSAT 2019 · Mathematics

The distance between the origin and the normal to curve $y = e^{2 x} + x^{2}$ at $x = 0$ is

A
$\frac{2}{\sqrt{3}}$
Correct answer
B
$\frac{2}{\sqrt{5}}$
C
$\frac{1}{\sqrt{3}}$
D
$\frac{1}{\sqrt{5}}$

Solution

We have,

$y = e^{2 x} + x^{2}$

at $x = 0, y = 1$

$\frac{d y}{d x} = 2 e^{2 x} + 2 x$

${(\frac{d y}{d x})}_{x = 0} = 2$

$∴$ Equation of normal at $(0, 1)$ is

$y - 1 = - \frac{1}{2} (x)$

$\Rightarrow x + 2 y - 2 = 0$

$∴$ Distance from $(0, 0)$ from normal $= \frac{2}{\sqrt{5}}$

Hence, the correct answer is option: 'B'.

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