Question

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

Answer 1

See explanation and graph.

Explanation:

Upon using

`( x, y ) = r ( cos theta, sin theta ) and r = sqrt ( x^2 + y^2 ) >= 0`,

`2 = ( x + 7 y )^2 - 12 x` converts to

`r^2 (cos theta + 7 sin theta )^2 -12 r cos theta - 2 = 0`. So,

`0 <= r =`

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

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

/`( 2 (cos theta + 7 sin theta )^2)`

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

`(y -m x - c )^2 = k (mx + y - c' )` to read immediately

Axis: `y - mx - c = 0`

Tangent at the vertex: .`mx + 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}