Questions: The ellipse x^2/5^2 + y^2/6^2 = 1 can be drawn with parametric equations where x(t) is written in the form x(t) = r cos(t) with r = and y(t) =

Transcript text: The ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{6^{2}}=1$ can be drawn with parametric equations where $x(t)$ is written in the form $x(t)=r \cos (t)$ with $r=$ $\square$ and $y(t)=$ $\square$

Solution

Solution Steps

Hint

To find the parametric equations for an ellipse, use the semi-major and semi-minor axes as coefficients for the cosine and sine functions, respectively, with the parameter representing an angle. This approach allows you to express the x and y coordinates in terms of a single variable, which can then be used to plot the ellipse.

Step 1: Identify the Standard Form of the Ellipse

The given equation of the ellipse is: \[ \frac{x^{2}}{5^{2}}+\frac{y^{2}}{6^{2}}=1 \] This is in the standard form of an ellipse: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where $a = 5$ and $b = 6$.

Step 2: Parametric Equations for the Ellipse

The parametric equations for an ellipse in the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are: \[ x(t) = a \cos(t) \] \[ y(t) = b \sin(t) \]

Step 3: Substitute the Values of $a$ and $b$

Given $a = 5$ and $b = 6$, we substitute these values into the parametric equations: \[ x(t) = 5 \cos(t) \] \[ y(t) = 6 \sin(t) \]