Parametric equation of Ellipse - Practice Questions & MCQ

Parametric equation of Ellipse:

The equations $\mathrm{x}=\mathrm{a} \cos \theta, \mathrm{y}=\mathrm{b} \sin \theta$ are called the parametric equation of the ellipse.

The circle described on the major axis as the diameter is called the auxiliary circle.

Equation of Ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Then, equation of auxiliary circle is $x^2+y^2=a^2$ (As $A A^{\prime}$ is Diameter)

Let $\mathrm{P}(\mathrm{x}, \mathrm{y})$ be a point on the ellipse
Draw PN perpendicular to the major axis and PN to meet the auxiliary circle at Q.
Let $\angle \mathrm{ACQ}$ be $\theta$ (This angle is also known as Eccentric Angle). Hence, the parametric equation of circle at point Q (a $\cos \Theta, \mathrm{a} \sin \Theta$ ).
Thus, P has x -coordinate as a $\cos \Theta$

As P lies on the ellipse

$
\frac{\mathrm{a}^2 \cos ^2 \theta}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1 \Rightarrow \mathrm{y}= \pm \mathrm{b} \sin \theta
$
Hence, Point P is $(\mathrm{a} \cos \theta, \mathrm{b} \sin \theta)$