I'm working on solving the equation for "r". This equation got way out of hand, and I'm not sure where to continue from here. The original equation is ( x-h / a )^2 + ( y-k / b )^2 = 1.
What I'm trying to do is compute the radius of an ellipse given an angle Theta, and the ellipse center (h,k). Where the radius line still starts at (0, 0).
I started by replacing x and y with r * cos(theta) and r * sin(theta) respectively. From there, I squared the numerator, and denominator as required by the equation which left me with an expanded equation that I would like to be able to solve for "r".
Here is a link to the work I've do so far:
On the left hand side of the image (separated by green line) is the solution for the same equation without the "-h" and "-k" parameters. And on the right is my attempt to solve for "r" with the "-h" and "-k" parameters.
I'm stuck at the spot on the right in trying to solve for "r" any assistance is much appreciated. I'm assuming there is some technique for simplifying the equation that I may have forgotten, but I'm not sure what to search for.
Thanks.
$$ x=h+a \cos (\theta )$$ and $$y=k+b \sin (\theta )$$
$$r^2= x^2+y^2 = (h+a \cos (\theta ))^2 +(k+b \sin (\theta ))^2$$ and
$$r= $$
$$\sqrt {(h+a \cos (\theta ))^2 +(k+b \sin (\theta ))^2}$$