# If the tangent to the curve,

Question:

If the tangent to the curve, $y=e^{x}$ at a point $\left(c, e^{c}\right)$ and the normal to the parabola, $y^{2}=4 x$ at the point $(1,2)$ intersect at the same point on the $x$-axis, then the value of $c$ is

Solution:

For $(1,2)$ of $y^{2}=4 x \Rightarrow t=1, a=1$

Equation of normal to the parabola

$\Rightarrow t x+y=2 a t+a t^{3}$

$\Rightarrow x+y=3$ intersect $x$-axis at $(3,0)$

$y=e^{x} \Rightarrow \frac{d y}{d x}=e^{x}$

Equation of tangent to the curve

$\Rightarrow y-e^{c}=e^{c}(x-c)$

$\because$ Tangent to the curve and normal to the parabola

intersect at same point.

$\therefore 0-e^{c}=e^{c}(3-c) \Rightarrow c=4$