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Particular Solution of a Differential Equation

If point Ph,k...

Question

If point $P (h, k)$ on the curve $y = x + ln x$ is at the shortest distance from straight line $y = 2 x + 3.$ Then the value $(h + k)$ is

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Solution

For shortest distance, slope of tangent on the curve will be equal to slope of line.
Now, first curve $: y = x + ln x$
Differentiating both sides w.r.t. $x,$
$\Rightarrow y^{'} = 1 + \frac{1}{x}$
At point $P (h, k), y^{'} = 1 + \frac{1}{h}$
$\Rightarrow {(\frac{d y}{d x})}_{c_{1}} = {(\frac{d y}{d x})}_{c_{2}}$
$\Rightarrow 1 + \frac{1}{h} = 2$
$∴ h = 1,$ and $(h, k)$ satisfies $y = x + ln x$
$\Rightarrow k = 1$
$∴ h + k = 2$

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Particular Solution of a Differential Equation

Standard XII Mathematics

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