If the tangent to the curve, $\mathrm{y}=\mathrm{e}^{\mathrm{x}}$ at a point ( $\mathrm{c}, \mathrm{e}^{\mathrm{c}}$ ) 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 ______.

JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Application of Derivatives

If the tangent to the curve,

y = e^{x}

at a point (

c, e^{c}

) 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 ______.

Answer: 4

The normal to

y^{2} = 4 x

(1, 2)

, where

a = 1

and the parameter

t = 1

, is

t x + y = 2 a t + a t^{3}

, that is

x + y = 3

, meeting the

x

-axis at

(3, 0)

. The tangent to

y = e^{x}

(c, e^{c})

has slope

e^{c}

y - e^{c} = e^{c} (x - c)

. For it to pass through

(3, 0)

- e^{c} = e^{c} (3 - c)

, so

- 1 = 3 - c

and

c = 4

JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Integer answer