Problem 46 - Precalculus

Question

Sketch (if possible) the graph of the degenerate conic. $$y^{2}-25 x^{2}=0$$

Step-by-Step Solution

Answer

The graph of \(y^{2} - 25x^{2} = 0\) is an X shape formed by the two intersecting lines \(y = 5x\) and \(y = -5x\).

Step 1: Rewrite the Equation

Firstly, rewrite the equation as \(y^{2} = 25x^{2}\). This is done by adding \(25x^{2}\) to both sides to isolate \(y^{2}\) on one side.

Step 2: Simplify the Equation

Next, the equation is simplified further by taking a square root of both sides, which results in \(y = \pm 5x\). This gives two separate equations \(y = 5x\) and \(y = -5x\), which will correspond to two intersecting lines on the graph.

Step 3: Sketch the Graph

Now, plot the two lines \(y = 5x\) and \(y = -5x\). Begin with the x and y values of 0, and then use the slope of 5 (or -5) to determine the next points. These two lines intersect at the origin, creating an X-shape. This is the graph of the given degenerate conic equation.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped right circular cone. They include circles, ellipses, parabolas, and hyperbolas, depending on the angle of the intersecting plane relative to the cone's surface. When the intersection occurs such that the plane passes through the apex of the cone, the result is a degenerate conic. In this case, rather than a smooth curve, you may get a single point, a line, or a pair of intersecting lines.

The equation provided, y^2 - 25x^2 = 0, represents a degenerate conic because it simplifies to a pair of lines. Such degenerations can be considered special cases within the family of conic sections where the usual smooth curves collapse into simpler geometric forms. The implications for graphing are that instead of plotting a curved path, one must consider these special cases that lead to linear graphs.

Graphing Equations

Graphing equations is a visual way of representing mathematical expressions. It can give students a more intuitive understanding of the relationship between variables. For linear equations, like the ones we obtain from the degenerate conic y^2 = 25x^2, graphing involves plotting lines on a coordinate system. In this example, after simplifying, we have two separate linear equations, y = 5x and y = -5x.

To graph these lines, one typically starts by finding the y-intercept and slope. However, here both lines pass through the origin, giving us a y-intercept of 0. The slope is the coefficient of x, which is 5 for one line and -5 for the other. These slopes tell us that for every unit increase in x, y increases by 5 units for the first line and decreases by 5 units for the second, thus creating an X-shape when the lines are graphed.

Simplifying Equations

Simplifying equations is essential for making complex algebra manageable and for revealing the underlying structure of mathematical expressions. In the context of the degenerate conic y^2 - 25x^2 = 0, simplification starts by realizing that the equation resembles a difference of squares, which factors as (y - 5x)(y + 5x) = 0. From there, you can see that the equation can split into two simpler linear equations: y = 5x and y = -5x.

By simplifying the equation as done in the solution steps, we discover that instead of representing a single geometric form, it represents two intersecting lines. This process of simplifying helps students move from the general form of a conic section to the specific forms that are easier to work with, whether for further algebraic manipulations or for graphing purposes.