Question

If point $(\alpha ,\beta )$ lies on the curve $y=\ln x$, is at the shortest distance from point $(2,3),$ then which of the following is true

• A. ${\alpha }^2-2\alpha +\beta =3$
• B. ${\alpha }^2-\alpha +\beta =3$
• C. ${\alpha }^2+2\alpha +\beta =3$
• D. ${\alpha }^2+\alpha +\beta =3$

Answer 1

The correct option is A ${\alpha }^2-2\alpha +\beta =3$

Given curve, $y=\ln x$
Slope of tangent at $(\alpha ,\beta )$
$={\frac{dy}{dx}}_{(\alpha ,\beta )}=\frac{1}{\alpha }$
Since $(2,3)$ is at the shortest distance from $(\alpha ,\beta )$
$\therefore$ Normal at $(\alpha ,\beta )$ passes through $(2,3)$
$\Rightarrow$ Slope of normal joining $(\alpha ,\beta )$ and $(2,3)$ $=\frac{\beta -3}{\alpha -2}$
Now, $\frac{1}{\alpha }\cdot \frac{\beta -3}{\alpha -2}=-1$
$\Rightarrow {\alpha }^2-2\alpha +\beta =3$