Problem 2 - Algebra 2

Question

Write an equation of an ellipse with the given characteristics. Check your answers. center \((5,3),\) vertical major axis of length \(12,\) minor axis of length 8

Step-by-Step Solution

Answer

The equation of the ellipse with center at (5, 3), with vertical major axis of length 12 and minor axis of length 8 is \(\frac{(x-5)^2}{16} + \frac{(y-3)^2}{36} = 1

Step 1: Identify the required variables from the problem

From the problem, we can identify the following variables:\n Center, \(h = 5, k = 3\)\n Length of major axis, \(2a = 12\) which implies \(a = 6\)\n Length of minor axis, \(2b = 8\) which implies \(b = 4.

Step 2: Substitute the variables into the equation of an ellipse with a vertical major axis

Plugging in the values of the variables into the equation of an ellipse with a vertical major axis \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\), gives us \(\frac{(x-5)^2}{4^2} + \frac{(y-3)^2}{6^2} = 1.\)

Step 3: Simplify the Equation

We can simplify the equation further to \(\frac{(x-5)^2}{16} + \frac{(y-3)^2}{36} = 1\) which is our final equation of the ellipse.

Key Concepts

Conic Sections

Conic sections are shapes formed by the intersection of a plane with a double cone. These shapes include circles, ellipses, parabolas, and hyperbolas.
Ellipses, specifically, result when the plane cuts through the cone at an angle to the base, not parallel and not perpendicular to the cone’s axis.
Understanding these shapes is essential because they appear frequently in geometry and various applications like orbital paths of planets.

• Circles are a special type of ellipse with equal major and minor axes.
• Ellipses have two focal points and the total distance from these points to any point on the ellipse is constant.
• Parabolas and hyperbolas are formed through different angles of intersection.

Recognizing the different conic sections is the first step in understanding their respective equations and characteristics.

Major and Minor Axes

In an ellipse, the terms major and minor axes refer to the longest and shortest diameters of the ellipse. The major axis is always longer than the minor axis.
These axes intersect at the center of the ellipse, and are perpendicular to each other.

• Major Axis: The longest line that can be drawn through the center of the ellipse.
• Minor Axis: The shortest line that spans the center of the ellipse.

In the problem given, the major axis is vertical, and it measures 12 units, meaning the semi-major axis is half of this length, which is 6 units.
The minor axis measures 8 units, resulting in a semi-minor axis of 4 units. The orientation of the axes is crucial for determining the standard form of the ellipse equation.

Standard Form of an Ellipse

The standard form of an ellipse's equation depends on the orientation of its axes. For an ellipse centered at \(h, k\), with a vertical major axis, the equation is:

• \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\)

When the major axis is horizontal, the positions of \(a^2\) and \(b^2\) are swapped:

• \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)

In our exercise, since the major axis is vertical, we use the first form. By substituting the known values:
\(h = 5\), \(k = 3\), \(a = 6\), and \(b = 4\), the equation becomes:
\(\frac{(x-5)^2}{16} + \frac{(y-3)^2}{36} = 1\).
This is the elliptical equation that fits the described parameters.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows the representation and analysis of geometric figures using a coordinate plane.
It is essential for solving problems involving shapes like ellipses as it combines algebra and geometry.

• It involves graphing and analyzing equations to derive insights about geometric figures.
• Using coordinates helps to describe positions and movements uniquely.
• It enables precise calculations and visualizations in mathematics.

In the context of ellipses, coordinate geometry allows us to find the ellipse's equation given its geometric properties such as center, axes lengths, and orientation.
This approach is integral for learners to transition from abstract shapes to concrete calculations, enhancing their spatial understanding and problem-solving skills.