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Q25.

Expert-verifiedFound in: Page 593

Book edition
7th Edition

Author(s)
James Stewart, Lothar Redlin, Saleem Watson

Pages
948 pages

ISBN
9781337067508

**Rectangular Coordinates to Polar Coordinates. A point is graphed in rectangular form. Find polar coordinates for the point, with $r>0$and $0<\theta <2\pi $**

The polar coordinates are**$3\sqrt{2},\frac{3\pi}{4}$**

A point in the rectangular coordinate system

Four points match the polar coordinates of a point. The radius of each point is four units. Their angles are the only thing that separates them. As a result, find the point that has the same angle as the specified point..

The coordinates of the point are. So,$x=-3$ and $y=3$. Now convert it into polar coordinates.

Use the formulae as-

$\begin{array}{l}{r}^{2}={x}^{2}+{y}^{2}\\ \mathrm{tan}\theta =\frac{y}{x}\end{array}$

$\begin{array}{l}\Rightarrow {r}^{2}={x}^{2}+{y}^{2}\\ \Rightarrow {r}^{2}\text{Invalid <m:msup> element}\text{Invalid <m:msup> element}=+\\ \Rightarrow {r}^{2}=18\\ \Rightarrow r=\pm \sqrt{18}or\pm 3\sqrt{2}\end{array}$

But they had given,$r>0$

Therefore, $r=+3\sqrt{2}$

The value of angle can be determined as-

$\begin{array}{l}\Rightarrow \mathrm{tan}\theta =\frac{y}{x}\\ \Rightarrow \mathrm{tan}\theta =\frac{3}{-3}\\ \Rightarrow \mathrm{tan}\theta =-1\\ \Rightarrow \theta ={\mathrm{tan}}^{-1}(-1)\\ \Rightarrow \theta =\frac{3\pi}{4},\frac{7\pi}{4}\end{array}$

But in the question,$0<\theta <2\pi $ . Therefore $\theta =\frac{3\pi}{4}$

Polar coordinates of the point are $3\sqrt{2},\frac{3\pi}{4}$

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