Question

The distance between the points $(2,\frac{\pi }{2}),(5,{\tan }^{-1}\frac{4}{3})$ is

• A. $2$
• B. $3$
• C. $\sqrt{13}$
• D. $13$

Answer 1

Answer: (C)

$\sqrt{13}$

Converting from polar to rectangular form.

$A\equiv (2,\frac{\pi }{2})$

Here, $r=2$ and ${\theta }_1=\frac{\pi }{2}$

Hence, $x_1=r\cos {\theta }_1$ and $y_1=r\sin {\theta }_1$

$\therefore x_1=0$ and $y_1=2$

So, $(2,\frac{\pi }{2})\equiv (0,2)$

$B\equiv (5,{\tan }^{-1}\frac{4}{3})$

Here, $r=5$ and $\tan {\theta }_2=\frac{4}{3}$

$\therefore \sin {\theta }_2=\frac{4}{5}$ and $\cos {\theta }_2=\frac{3}{5}$

Hence, $x_2=r\cos {\theta }_2$ and $y_2=r\sin {\theta }_2$

$\therefore x_2=3$ and $y_2=4$

So, $(5,{\tan }^{-1}\frac{4}{3})\equiv (3,4)$

By distance formula,

$AB=\sqrt{(3-0{)}^2+(4-2{)}^2}$

$AB=\sqrt{13}$