Problem 9 - Intermediate Algebra

Question

Find the \(x\) -and the \(y\) -intercepts of the graph of \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)

Step-by-Step Solution

Answer

The x-intercepts are \((a, 0)\) and \((-a, 0)\). The y-intercepts are \((0, b)\) and \((0, -b)\).

Step 1: Understand the equation format

The given equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) describes an ellipse centered at the origin. To find the intercepts, we will set \(y = 0\) to find the \(x\)-intercept and set \(x = 0\) to find the \(y\)-intercept.

Step 2: Find the x-intercept

To find the \(x\)-intercept, set \(y = 0\) in the equation: \[\frac{x^{2}}{a^{2}}+\frac{0^{2}}{b^{2}}=1\]. Simplifying, we get \(\frac{x^{2}}{a^{2}}=1\). Solving for \(x\), we multiply both sides by \(a^{2}\) to get \(x^{2} = a^{2}\), which gives \(x = \pm a\). Therefore, the \(x\)-intercepts are \((a, 0)\) and \((-a, 0)\).

Step 3: Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation: \[\frac{0^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\]. Simplifying, we get \(\frac{y^{2}}{b^{2}}=1\). Solving for \(y\), we multiply both sides by \(b^{2}\) to get \(y^{2} = b^{2}\), which gives \(y = \pm b\). Therefore, the \(y\)-intercepts are \((0, b)\) and \((0, -b)\).

Key Concepts

Ellipse Equation

An ellipse is a fascinating shape that resembles a stretched circle. It is described mathematically by the equation \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \). This equation is known as the standard form of an ellipse centered at the origin of a coordinate plane. Here, \(a\) and \(b\) are the semi-major and semi-minor axes, respectively, determining the ellipse's size and shape.

In this equation:

• \(x\) and \(y\) are the coordinates of any point on the ellipse.
• \(a\) is the distance from the center of the ellipse to its vertex along the x-axis.
• \(b\) is the distance from the center to the vertex along the y-axis.

These parameters \(a\) and \(b\) must be positive and are crucial in learning about the ellipse's intercepts and how they stretch in different directions. Understanding these parts of the ellipse equation makes finding the intercepts a straightforward task.

X-Intercepts

Finding the x-intercepts of an ellipse is simple once you have the equation. X-intercepts occur where the ellipse crosses the x-axis, which means the value of \(y\) is zero at these points. To find them, set \(y = 0\) in the ellipse equation \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \). This simplifies to \( \frac{x^{2}}{a^{2}} = 1 \).

By solving this equation, you multiply both sides by \(a^{2}\) to get \(x^{2} = a^{2}\). Taking the square root of both sides results in \(x = \pm a\). Therefore, the ellipse has two x-intercepts, known as:

• \((a, 0)\)
• \((-a, 0)\)

These points are where the ellipse's shape reaches its furthest extent along the x-axis.

Y-Intercepts

Just as with the x-intercepts, y-intercepts are found where the ellipse crosses the y-axis, meaning the x-coordinate is zero at these points. To discover the y-intercepts, set \(x = 0\) in the same ellipse equation: \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \). This simplifies to \( \frac{y^{2}}{b^{2}} = 1 \).

Solving for \(y\), multiply both sides by \(b^{2}\) to arrive at \(y^{2} = b^{2}\). Taking the square root of both sides gives us \(y = \pm b\). Hence, the ellipse has two y-intercepts, depicted as:

• \((0, b)\)
• \((0, -b)\)

These are the points along the y-axis where the ellipse reaches its highest and lowest positions. Understanding these intercepts is an important part of studying the geometry of ellipses and how they interact with the coordinate plane.