Question

What are represented by the equation $r(4-3{\sin }^2\theta )=8a\cos \theta .$

Answer 1

Here: $r=\sqrt{x^2+y^2}$, $x=rcos\theta$ & $y=rsin\theta$

Given equation: $r(4-3si{n}^2\theta )=8acos\theta$

$\Rightarrow r(4-3{\left(\frac{y}{r}\right)}^2)=8a\frac{x}{r}$

$\Rightarrow r^2.\frac{4r^2-3y^2}{r^2}=8ax$

$\Rightarrow 4(\sqrt{x^2+y^2}{)}^2-3y^2=8ax$

$\Rightarrow 4x^2+4y^2-3y^2=8ax$

$\Rightarrow 4x^2-8ax+y^2=0$

$\Rightarrow (2x{)}^2-2.2x.2a+(2a{)}^2+y^2=4a^2$

$\Rightarrow (2x-2a{)}^2+y^2=4a^2$

$\Rightarrow \frac{(x-a{)}^2}{a^2}+\frac{y^2}{4a^2}=1$

This equation is similar to a standard ellipse.

Therefore this represents an $ellipse$.