Question

How do you combine the parametric equations into one equation relating y to x given $x=4\cos t$ and $y=9\sin t$?

Answer 1

Hint: We explain the number of ways position of a point or equation can be expressed in different forms. We also explain the ways the representation works for polar and cartesian form. We form the given equation $x=4\cos t$ and $y=9\sin t$ in the values of the formula ${{\sin }^{2}}t+{{\cos }^{2}}t=1$.

Complete step by step answer:
There are always two ways to represent any point equation in our general 2-D and 3-D surfaces. One being the polar form and the other one being the cartesian form. The other name of the cartesian form is rectangular form.
In case of polar form, we use the distance and the angle from the origin to get the position of the point or curve.
The given equations $x=4\cos t$ and $y=9\sin t$ are representations of the polar form. r represents the distance and $\theta $ represents the angle.
In case of rectangular form, we use the coordinates from the origin to get the position of the point or curve. For two dimensional things we have X-Y and for three dimensional things we have X-Y-Z. We take the perpendicular distances from the axes.
We need to convert the given equations $x=4\cos t$ and $y=9\sin t$ into the rectangular form.
The relation between these two forms in two-dimensional is ${{\sin }^{2}}t+{{\cos }^{2}}t=1$.
From the relation $x=4\cos t$ we get $\cos t=\dfrac{x}{4}$.
From the relation $y=9\sin t$ we get $\sin t=\dfrac{y}{9}$.
We now replace the value of $\sin t$ and $\cos t$ in the equation ${{\sin }^{2}}t+{{\cos }^{2}}t=1$ to get
\[\begin{align}
  & {{\sin }^{2}}t+{{\cos }^{2}}t=1 \\
 & \Rightarrow {{\left( \dfrac{y}{9} \right)}^{2}}+{{\left( \dfrac{x}{4} \right)}^{2}}=1 \\
 & \Rightarrow \dfrac{{{x}^{2}}}{16}+\dfrac{{{y}^{2}}}{81}=1 \\
\end{align}\]
The equation is an equation of the ellipse.
This is the rectangular form of $x=4\cos t$ and $y=9\sin t$.

Note: In case of points for cartesian form we use x and y coordinates as $\left( x,y \right)$ to express their position in the cartesian plane. The distance from origin is $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$. This r represents the distance in polar form. In the case of an ellipse the distance is measured with respect to the focus.