RELATIONSHIPS BETWEEN POLAR AND CARTESIAN COORDINATES

RELATIONSHIPS BETWEEN POLAR AND CARTESIAN COORDINATES


C
Asked by 1 month ago
10 million points

Let $P$ be any point and let $OP=r$, where $O$ is the origin and let $OP$ make an angle $\theta$ with the axis, then $r$ and $\theta$ are called Polar coordinates of the point $P$, and the coordinates may be written as $(r,\theta)$.

Find the Cartesian equations of

(a) $r=a(1+2\cos\theta)$,

RELATIONSHIPS BETWEEN
chegendungu

1 Answer

C
Answered by 1 month ago
10 million points

(a) $r=a(1+2\cos\theta)$

[The $\cos\theta$ suggests the relation $x=r\cos\theta$, so multiply through by $r$.]

$\therefore r^2=a(r+2r\cos\theta)$

$\therefore x^2+y^2=a(\sqrt{(x^2+y^2)}+2x)$

$\therefore x^2+y^2-2ax=a\sqrt{(x^2+y^2)}$

Therefore the Cartesian equation of $r=a(1+2\cos\theta) is

$(x^2+y^2-2ax)^2=a^2(x^2+y^2)$

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Asked: 1 month ago

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