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Diameter : Hyperbola

If the tangen...

Question

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

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Solution

For $(1, 2)$ of $y^{2} = 4 x$
$\Rightarrow t = 1, a = 1$
Equation of normal $\Rightarrow t x + y = 2 a t + a t^{3}$
$\Rightarrow x + y = 3$ intersect $x$ -axis at $(3, 0)$

$y = e^{x}$
$\Rightarrow \frac{d y}{d x} = e^{x}$
Tangent at point $(c, e^{c})$ is
$\Rightarrow y - e^{c} = e^{c} (x - c)$
at $(3, 0) \Rightarrow 0 - e^{c} = e^{c} (3 - c)$
$∴ c = 4$

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