How do you convert 2=(x+7y)^2-12x into polar form?

Jul 18, 2018

See explanation and graph.

Explanation:

Upon using

$\left(x , y\right) = r \left(\cos \theta , \sin \theta\right) \mathmr{and} r = \sqrt{{x}^{2} + {y}^{2}} \ge 0$,

$2 = {\left(x + 7 y\right)}^{2} - 12 x$ converts to

${r}^{2} {\left(\cos \theta + 7 \sin \theta\right)}^{2} - 12 r \cos \theta - 2 = 0$. So,

$0 \le r =$

(12 cos theta + sqrt (144 cos^2theta

+ 4 (cos theta + 7 sin theta)^2)

/$\left(2 {\left(\cos \theta + 7 \sin \theta\right)}^{2}\right)$

The purpose of this conversion is not known.

Analysis of the given Cartesian equation that represents a

parabola is relatively quite easier.

It is known that, if the second degree terms form a perfect

square, a second degree equation represents a parabola.

I know that the readers would like what follows.

The axis and the tangent at the vertex are at right angles. So, the

equation of any parabola can be converted to the form

${\left(y - m x - c\right)}^{2} = k \left(m x + y - c '\right)$ to read immediately

Axis: $y - m x - c = 0$

Tangent at the vertex: .$m x + y - c ' = 0$

I have worked this out here. It is

(x + 7y - 0.12 )^2 = 1.68 ( 7x - y ) + 2.0144. See graph.
graph{((x+7y)^2-12x-2)(x + 7y - 0.12 )(1.68 ( 7x - y ) + 2.0144)=0}