Question

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

Answer 1

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. $x,$
$\Rightarrow y^{\prime }=1+\frac{1}{x}$
At point $P(h,k),y^{\prime }=1+\frac{1}{h}$
$\Rightarrow {\left(\frac{dy}{dx}\right)}_{c_1}={\left(\frac{dy}{dx}\right)}_{c_2}$
$\Rightarrow 1+\frac{1}{h}=2$
$\therefore h=1,$ and$(h,k)$ satisfies $y=x+\ln x$
$\Rightarrow k=1$
$\therefore h+k=2$