Question

The shortest distance between the point $\left(\frac{3}{2} ,0\right) $ and the curve $ y = \sqrt{x}, \left(x > 0\right)$ is :

• A. $\frac{\sqrt{5}}{2}$
• B. $\frac{5}{4}$
• C. $\frac{3}{2}$
• D. $\frac{\sqrt{3}}{2}$

Answer 1

Answer: (A)

Let points $\left(\frac{3}{2}, 0\right),\left( t ^{2}, t \right), t >0$
Distance $=\sqrt{ t ^{2}+\left( t ^{2}-\frac{3}{2}\right)^{2}}$
$=\sqrt{ t ^{4}-2 t ^{2}+\frac{9}{4}}=\sqrt{\left( t ^{2}-1\right)^{2}+\frac{5}{4}}$
So minimum distance is $\sqrt{\frac{5}{4}}=\frac{\sqrt{5}}{2}$