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Eccentricity of Ellipse

If point P0,1...

Question

If point $P (0, 1)$ on the curve $y = x + e^{sin x}$ is at the shortest distance from $y = m x + c .$ Then the value of $m^{2}$ is

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Solution

For shortest distance, slope of tangent on the curve will be equal to the slope of line.
Now, first curve : $y = x + e^{sin x}$
Differentiating both sides w.r.t. $x,$
$\Rightarrow y^{'} = 1 + e^{sin x} \cdot cos x$
At point $(0, 1), y^{'} = 2$
$\Rightarrow {(\frac{d y}{d x})}_{c_{1}} = {(\frac{d y}{d x})}_{c_{2}}$
$\Rightarrow 2 = m$
$∴ m^{2} = 4$

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Eccentricity of Ellipse

Standard XII Mathematics

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