91Ó°ÊÓ

An ellipse is a shape that looks like a stretched out circle. It has two main parts:
the longer axis, called the major axis, and the shorter one, called the minor axis.
For our ellipse, based on the equations given, the semi-major axis is 3 and the semi-minor axis is 2.
The center of the ellipse is at the origin (0,0).

The general equation of an ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Where 'a' is the length of the semi-major axis, and 'b' is the length of the semi-minor axis.
With our specific example, we have:
\[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \]
This indicates an ellipse centered at the origin with lengths 2 and 3 for the axes.
Ellipses have many interesting properties, like the fact that the sum of the distances from any point on an ellipse to two special points (called foci) is always the same.

In parametric equations, a variable (often 't') defines both 'x' and 'y'.
The goal here is to eliminate 't' and find a direct relationship between 'x' and 'y'.
This relationship is called a rectangular equation because it doesn't rely on a third variable.

To eliminate 't', we use trigonometric identities.
Given:
\[ x(t) = 2 \cos t \]
and
\[ y(t) = 3 \sin t \]
We can solve these equations for cos(t) and sin(t) in terms of x and y.
So, by isolating cos(t) and sin(t), and using the Pythagorean identity \( \cos^2 t + \sin^2 t = 1 \)
we get a new equation that relates \( x \) and \( y \) directly:

\[ \left( \frac{x}{2} \right)^2 + \left( \frac{y}{3} \right)^2 = 1 \]
Upon simplifying, this gives:
\[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \]

Trigonometric identities are key relationships in trigonometry.
They define the relationships between various trigonometric functions, like sine, cosine, tangent, and more.
One of the most useful identities is the Pythagorean identity:

\[ \cos^2 t + \sin^2 t = 1 \]
This identity helps us eliminate the parameter 't' from parametric equations.
In our case:
Given:
\[ x = 2 \cos t \quad \text{and} \quad y = 3 \sin t \]
We solve for \( \cos t \) and \( \sin t \):
\[ \cos t = \frac{x}{2} \quad \text{and} \quad \sin t = \frac{y}{3} \]
Substituting these into the Pythagorean identity:

\[ \left( \frac{x}{2} \right)^2 + \left( \frac{y}{3} \right)^2 = 1 \]
simplifies to:
\[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \]
This becomes the rectangular equation of our ellipse.

Graphing curves is about plotting points that satisfy an equation.
In parametric equations, you plot (x(t), y(t)) for a range of values of 't'.
Given the range \(0 \leq t \leq 2\pi\), this means we plot these values as 't' changes from 0 to \(2\pi\).

For our equations:
\[ x(t) = 2 \cos t, \quad y(t) = 3 \sin t \]
When 't' changes from 0 to \(2\pi\), we get an ellipse.
In the rectangular form we found:
\[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \]
We can use this form to plot the ellipse precisely.

The graph has a semi-major axis of 3 and a semi-minor axis of 2.
It starts at point (2,0) when t=0 and traces counterclockwise back to this point as t goes from 0 to \(2\pi\).