# If the tangent to the curve, $y=e^{x}$ at a point $(c,e^{c})$ and the normal to the parabola, $y^{2}=4x$ at the point $(1,2)$ intersect at the same point on the x-axis, then the value of $c$ is _______. Option: 1 2 Option: 2 4 Option: 3 -2 Option: 4 -4

\begin{aligned} &\begin{array}{l} y=e^{x} \Rightarrow \frac{d y}{d x}=e^{x} \\ m=\left(\frac{d y}{d x}\right)_{\left(c, e^{c}\right)}=e^{c} \end{array}\\ & \text { Tangent at }\left(\mathrm{c}, \mathrm{e}^{\mathrm{c}}\right)\\ &y-e^{c}=e^{c}(x-c) \end{aligned}

it intersects x-axis

\begin{aligned} &\text { Put } \quad y=0 \Rightarrow x=c-1\\ &\text { Now } y^{2}=4 x \Rightarrow \frac{d y}{d x}=\frac{2}{y} \Rightarrow\left(\frac{d y}{d x}\right)_{(1,2)}=1\\ &\Rightarrow \text { Slope of normal }=-1 \end{aligned}

Equation of normal y – 2 = – 1(x – 1)

x + y = 3 it intersect x-axis

\begin{aligned} &\text { Put } \mathrm{y}=0 \Rightarrow \mathrm{x}=3\\ &\text { Points are same }\\ &\Rightarrow x=c-1=3\\ &\Rightarrow c=4 \end{aligned}

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