Question

The parametric equations $x=2\sin t,y=3\cos t$define a(n)______.

Short answer

The parametric equations $x=2\sin t,y=3\cos t$define a(n) ellipse.

Step-by-step solution

Step 1. Given information.

The parametric equations$x=2\sin t,y=3\cos t$are given.

Step 2. Fill the blank.

The parametric equation of an ellipse is defined as;

$x=a\sin \left(t\right),y=b\cos \left(t\right)$

After comparing the above equations from the given equations.

We get, $a=2,b=3$

Thus, the equation of an ellipse is written as,

$$\begin{gathered} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \\ \Rightarrow \frac{x^2}{4}+\frac{y^2}{9}=1 \end{gathered}$$

The parametric equation defines a(n) ellipse because an ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.