Problem 61 - Precalculus

Question

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$4 x^{2}+3 y^{2}+8 x-24 y+51=0$$

Step-by-Step Solution

Answer

The given graph of the equation represents an ellipse.

Step 1 Revise the equation

The given equation is \(4x^{2} + 3y^{2} + 8x - 24y + 51 = 0\). We need to rewrite it in order to classify it.

Step 2 Regrouping the equation

Rearrange the equation terms into groups with identical variable power. The equation can now be written as: \(4(x^{2} + 2x) + 3(y^{2} - 8y) + 51 = 0\)

Step 3 Completing the square

We complete the square for both x and y terms. To complete the square, we take half the coefficient of the x or y term and square it, then add and subtract it inside the bracket. The equation will now be: \(4[(x^{2} + 2x + 1) - 1] + 3[(y^{2} - 8y + 16) - 16] + 51 = 0\)

Step 4 Simplify the equation

Now simplify the equation, each bracket will be the square of a binomial. The equation now will be: \(4[(x+1)^{2} - 1] + 3[(y-4)^{2} - 16] + 51 = 0\). If we rewrite this we get: \(4(x+1)^{2} + 3(y-4)^{2} = 4 + 48 - 51 = 1\)

Step 5 Identifying the conic section

In the form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), if both coefficients a and b are non zero and equal, it is a circle. If a or b is zero, it's a parabola. If a and b are both non zero but not equal, it is an ellipse. If there is a minus sign between two terms, it's a hyperbola. In our case, the standard form of the equation is in the format of an ellipse.

Key Concepts

Ellipse

An ellipse is a kind of conic section that emerges when a plane intersects a cone at an angle that is neither parallel nor perpendicular to the base. Imagine a stretched circle or an oval shape. Its key characteristics include two axes: the major axis, which is the longest path across the ellipse, and the minor axis, the shortest path through its center.

Ellipses hold two focal points inside them. Every point on the ellipse has a combined distance to these foci that remains constant. This property is important because it helps in determining the shape and size of an ellipse in mathematical terms.

• The major and minor axes determine the elongation of the ellipse.
• The foci are crucial for understanding the geometric nature of an ellipse.
• Ellipses are often seen in planetary orbits because they follow this path due to gravitational forces.

Understanding the basic properties of ellipses can help solve equations involving them!

Completing the Square

Completing the square is a mathematical technique used to simplify quadratic equations, and it plays a pivotal role in conic sections like ellipses. This method comes in handy to transform equations to their standard form, making them easier to analyze and understand.

In the context of an ellipse, completing the square involves grouping the terms of the equation, usually related to the same variable, and then performing the complete square process.

Here's a straightforward way to complete the square:

• First, identify the quadratic and linear terms for one variable, say, \(x^2 + bx\).
• Take half of the linear coefficient (\(b/2\)) and square it.
• Add and subtract this squared number inside the equation, allowing you to rewrite it as a squared binomial.

This transformation helps in highlighting the center and axes (lengths of semi-major and semi-minor) of the ellipse once everything is rearranged properly. It's not just a mechanical process, but a technique that reveals underlying geometric properties.

Equation of an Ellipse

The standard equation of an ellipse is an elegant representation that provides comprehensive insight into its geometric properties. When you see the form: \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\], it quickly tells you several things:

• \((h, k)\) is the center of the ellipse.
• \(a\) represents the semi-major axis's length, aligned with the x-axis if \(a > b\).
• \(b\) is the semi-minor axis's length, primarily oriented along the y-axis if \(b > a\).

The values of \(a^2\) and \(b^2\) determine how wide or tall the ellipse is; they're the key to understanding its shape. An important aspect of this formula is symmetry, where the parameters \(a\) and \(b\) create a perfect balance in how the ellipse is oriented around its center.

Transforming the equation of an ellipse to its standard form via methods like completing the square helps mathematicians and students to visualize and analyze the properties and chart the graph easily. This standard form captures the essence of ellipses, making them easier to work with in both theoretical and applied contexts.