Step-by-step explanation:
Given:
[tex]r^2(b^2\cos^2{\theta} + a^2\sin^2{\theta}) = a^2b^2[/tex]
or
[tex]b^2(r^2\cos^2{\theta}) + a^2(r^2\sin^2{\theta}) = a^2b^2[/tex]
Let [tex]x = r\cos{\theta}[/tex] and [tex]y = \sin{\theta}.[/tex] Substituting these into our given equation, we get
[tex]b^2x^2 + a^2y^2 = a^2b^2[/tex]
Dividing both sides by [tex]a^2b^2,[/tex] we get
[tex]\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1[/tex]
which we recognize as an equation for an ellipse.
