Chapter 8.4, Problem 34E

To determine

To find: the parametric equations for the ellipse.

Answer to Problem 34E

The parametric equations of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are $x=a\cos t$ and $y=b\sin t$ .

Explanation of Solution

Given:

  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Calculation:

Find the parametric equations of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .

On substituting $x=a\cos t$ and $y=b\sin t$ in $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ , we get

  $\frac{{\left(a\cos t\right)}^2}{a^2}+\frac{{\left(b\sin t\right)}^2}{b^2}=1$ .

On solving further, we get

  $\frac{a^2{\cos }^{2}t}{a^2}+\frac{b^2{\sin }^{2}t}{b^2}=1$ .

Therefore,

  ${\cos }^{2}t+{\sin }^{2}t=1$

Using ${\cos }^{2}t+{\sin }^{2}t=1$ , we get

  $1=1$ .

Therefore, left hand side of the equation is equal to the right hand side of the equation.

Thus, the parametric equations of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are $x=a\cos t$ and $y=b\sin t$ .

Conclusion:

Thus, the parametric equations of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are $x=a\cos t$ and $y=b\sin t$ .