Problem 1 - Algebra 2

Question

Write an equation of an ellipse in standard form with center at the origin and with the given vertex and co-vertex. $$ (4,0),(0,3) $$

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1\)

Step 1: Identify the values of 'a' and 'b'

Given the center of the ellipse at the origin, a vertex at (4,0) and a co-vertex at (0,3). Since the vertex is 4 units away from the center (origin), a = 4. And as the co-vertex is 3 units away from the center (origin), b = 3.

Step 2: Substitute the values of 'a' and 'b' in the standard equation of the ellipse

Now, we substitute the values of a and b into the standard equation of the ellipse: \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). We get: \(\frac{x^{2}}{4^{2}} + \frac{y^{2}}{3^{2}} = 1\)

Step 3: Simplify the equation

After simplification, we get the final equation of the ellipse as: \(\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1\)

Key Concepts

Standard Form of Ellipse

The standard form of an ellipse equation is essential for identifying the shape and position of an ellipse on a coordinate plane. An ellipse is essentially an elongated circle, defined by its semi-major and semi-minor axes. To write the equation in standard form, we use:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) for horizontal ellipses.
• \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \) for vertical ellipses.

Here, "a" represents the semi-major axis, which is the longer axis of the ellipse. "b" represents the semi-minor axis, the shorter one. The values \(a^2\) and \(b^2\) are found under the respective variables, \(x\) and \(y\), depending upon the orientation. It's important to notice that the equation is always equal to 1, which helps form the complete shape of the ellipse by maintaining its constant ratio.

Center at the Origin

The center of an ellipse is the midpoint around which the ellipse is symmetrically formed. When the center is at the origin, it means the ellipse is positioned at the point where both \(x\) and \(y\) coordinates are 0. This significantly simplifies the equation as it removes additional terms for horizontal and vertical shifts.An ellipse centered at the origin can be expressed in the simplest standard form without requiring any adjustments for a shifted center. Locating the center at (0, 0) allows us to directly establish relationships among the distance of the vertex and co-vertex from the center. By doing so, it ensures that the axes align perfectly along the coordinate plane, making calculations easier and intuitive.

Vertex and Co-vertex

The vertex and co-vertex of an ellipse are critical points that define how an ellipse stretches on the plane. The vertex is the furthest point on the ellipse from the center along the major axis, while the co-vertex is the furthest point along the minor axis. For an ellipse centered at the origin:

• If the vertex is (4,0), that means the major axis is horizontal, and "a" equals 4. Hence, the ellipse extends 4 units in the positive and negative x-direction.
• If the co-vertex is (0,3), this indicates the minor axis is vertical, and "b" equals 3. Thus, the ellipse extends 3 units upwards and downwards from the origin along the y-axis.

These measurements are crucial in forming the standard equation of the ellipse, as "a" and "b" directly determine the length and direction of the ellipse's axes. Understanding these positions helps in drawing and visualizing the ellipse accurately within the coordinate system.