The distance between the origin and the normal to curve y = e^ 2x +...

We have,

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

at $x = 0, y = 1$

$\dfrac{dy}{dx} = 2e^{2x} + 2x$

\left(\dfrac{dy}{dx}\right)x = 0 = 2$</p><p>$∴$Equation of normal at~$( 0, 1 )$is</p><p>$y - 1 = - \dfrac{1}{2} ( x )$</p><p>$⇒ x + 2y - 2 = 0$</p><p>$∴$Distance from$( 0, 0 )$from normal$= \dfrac{2}{5}