Problem 59 - Algebra 2

Question

Write an equation of an ellipse in standard form with center at the origin and with the given characteristics. focus \((0,3), y\) -intercept 5

Step-by-Step Solution

Answer

The standard form equation of the ellipse is \(\frac{x^2}{25} + \frac{y^2}{9} = 1\).

Step 1: Identify the Given Values

Start by noting the given focus $(0,3)$ and a point on the ellipse $(0,5)$. Since these are coordinates, they will be used to find the values of the semi-minor and semi-major axes.

Step 2: Calculate the Semi-Minor Axis

The distance from the center of the ellipse to the foci determines the semi-minor axis, denoted as \$b\$. This is given as the y-coordinate of the focus point, so the semi-minor axis \(b=3\).

Step 3: Calculate the Semi-Major Axis

The semi-major axis, represented as \$a\$, is the distance from the center of the ellipse to the furthest point on the ellipse (y-intercept). This is given as the y-coordinate of the point on the ellipse, so the semi-major axis \(a=5\).

Step 4: Construct the Standard Form Equation

With the center at the origin, the standard form equation of the ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Substitute \(a=5\) and \(b=3\) into the equation to get the final result.

Key Concepts

Conic Sections

Conic sections are fascinating and vital components of geometry. They are formed by the intersection of a plane and a double-napped cone.
There are four primary types of conic sections:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

These shapes have real-world applications in fields like engineering, astronomy, and physics. Among these, the ellipse is distinctive due to its oval shape, defined by two foci. The ellipse is not only visually appealing but also mathematically significant. It represents solutions to quadratic equations in two variables. Unlike circles, which are a special type of ellipse, ellipses have different lengths for their major and minor axes. This difference gives the ellipse its elongated shape.

Semi-Major Axis

The semi-major axis is an important parameter when dealing with ellipses. It is one half of the major axis, the longest diameter of the ellipse.
In the context of our problem, the semi-major axis is noted from the y-intercept, which is point (0,5). Since the center is at the origin, this means our semi-major axis, denoted as a, is 5. This length runs from the center to the furthest point around the elliptical path.
Understanding the semi-major axis helps in grasping the ellipse's size and orientation. An ellipse is broader along the axis associated with the semi-major, which distinctly separates it from the circular shape.

Semi-Minor Axis

Alongside the semi-major axis, the semi-minor axis is another critical component of an ellipse. This axis is the shorter diameter and is half the length of the minor axis.
In our exercise, the semi-minor axis is determined using the focus coordinate (0,3). Thus, the semi-minor axis, denoted as b, is 3. This measurement reflects the distance from the center of the ellipse to one of its narrowest points.
The lengths of both semi-major and semi-minor axes define the ellipse's shape completely, indicating how "stretched" or "compressed" it is.

Standard Form of Ellipse

The standard form of an ellipse equation provides a clear framework to express its structure mathematically. This form is written as:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]Here, a and b represent the semi-major and semi-minor axes, respectively. To understand the equation's layout, remember that the center of the ellipse in this form is at the origin (0,0).
In the specific problem, substituting a = 5 and b = 3 into the standard form of an ellipse yields:\[ \frac{x^2}{5^2} + \frac{y^2}{3^2} = 1 \] This equation offers a powerful tool for graphing the ellipse and understanding its properties, showing how its axes determine its geometry effectively. Standard form makes recognizing and working with ellipses much easier in mathematics.