Question

How do you convert `6=(5x+3y)^2-7x` into polar form?

Answer 1

`r^2(5cos theta + 3 sin theta )^2-7r cos theta-6= 0`

Explanation:

The second degree terms, in this second degree equation, form a

perfect square. As illustrated by the graph, the equation represents

a parabola..

The conversion formula is `(x, y)=r(cos theta, sin theta)`

Making these substitutions,

`r^2(5cos theta + 3 sin theta )^2-7r cos theta-6= 0`

I do not see any purpose in solving this quadratic in r, for r-explicit

form. It is yet another complicated form.

Interestingly, the polar equation of a parabola, referred to the focus

as pole r = 0, and the axis (away from the vertex ) as the initial line

`theta = 0` is

`(2a)/r=1+cos theta`, where a is the size of the parabola = the

distance between the focus ( pole ) and the vertex. The graph shows

which is which, in this transformed polar frame

graph{(5x+3y)^2-7x-6=0x^2 [-5, 5, -2.5, 2.5]}