Problem 57 - Intermediate Algebra

Question

The graph of equation is an ellipse. Determine which distance is longer, the distance between the \(x\)-intercepts or the distance between the \(y\) -intercepts. How much longer? \(4 x^{2}+y^{2}=16\)

Step-by-Step Solution

Answer

The distance between the y-intercepts is 4 units longer than between the x-intercepts.

Step 1: Rewrite the Equation in Standard Form

The given equation is \(4x^2 + y^2 = 16\). Divide every term by 16 to get the equation in the standard form of an ellipse: \(\frac{x^2}{4} + \frac{y^2}{16} = 1\). This shows that the ellipse is stretched along the y-axis.

Step 2: Calculate the X-Intercepts

To find the x-intercepts, set \(y = 0\) in the equation \(\frac{x^2}{4} + \frac{y^2}{16} = 1\). This gives \(\frac{x^2}{4} = 1\) or \(x^2 = 4\). Solving this gives \(x = 2\) and \(x = -2\). Thus, the x-intercepts are at \((-2, 0)\) and \((2, 0)\).

Step 3: Calculate the Distance Between X-Intercepts

The distance between the x-intercepts can be calculated using the formula for the distance between two points. Since the points are \((2, 0)\) and \((-2, 0)\), the distance is \(|2 - (-2)| = 4\).

Step 4: Calculate the Y-Intercepts

To find the y-intercepts, set \(x = 0\) in the equation \(\frac{x^2}{4} + \frac{y^2}{16} = 1\). This gives \(\frac{y^2}{16} = 1\) or \(y^2 = 16\). Solving this gives \(y = 4\) and \(y = -4\). Thus, the y-intercepts are at \((0, 4)\) and \((0, -4)\).

Step 5: Calculate the Distance Between Y-Intercepts

The distance between the y-intercepts is calculated similarly. Since the points are \((0, 4)\) and \((0, -4)\), the distance is \(|4 - (-4)| = 8\).

Step 6: Compare Distances and Calculate Difference

The distance between the x-intercepts is 4, and the distance between the y-intercepts is 8. To find how much longer the distance between the y-intercepts is, subtract the distances: \(8 - 4 = 4\).

Key Concepts

Understanding X-Intercepts of an Ellipse

Intercepts are the points where a curve crosses the axes. For x-intercepts, they occur where the graph intersects the x-axis, meaning at these points the y-value is zero.
To find the x-intercepts for an ellipse, you substitute zero for y in the ellipse's equation. For example, with the equation:

• \(\frac{x^2}{4} + \frac{y^2}{16} = 1\)

Substituting zero for y gives \(\frac{x^2}{4} = 1\), which means \(x^2 = 4\). Solving this results in two x-intercepts at \((2, 0)\) and \((-2, 0)\). These points reflect the horizontal stretch along the x-axis. The x-intercepts are crucial when analyzing the width or spread of an ellipse in relation to the x-axis.

Exploring Y-Intercepts of an Ellipse

The y-intercepts are similar to x-intercepts but occur where the graph cuts through the y-axis, with x being zero at these points.
To find y-intercepts, set x to zero in the standard form of the ellipse equation. In our ellipse example:

• \(\frac{x^2}{4} + \frac{y^2}{16} = 1\).

Rewriting it by substituting x = 0 gives \(\frac{y^2}{16} = 1\), leading to \(y^2 = 16\). Solving this results in the y-intercepts \((0, 4)\) and \((0, -4)\). These values highlight the elliptical stretch along the y-axis. Understanding these points helps in visualizing the height or vertical extent of the ellipse.

Applying the Distance Formula

The distance formula is vital in determining the distance between two points in a plane. It’s especially useful for calculating the distance between intercepts.
The formula is derived from the Pythagorean theorem and written as:

• \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

For horizontal distances like with the x-intercepts \((2, 0)\) and \((-2, 0)\), you just take the difference in x-values: \(|2 - (-2)| = 4\). Since the y-values are the same, the distance is only determined by x. For vertical distances, such as with y-intercepts \((0, 4)\) and \((0, -4)\), focus on the difference in y-values: \(|4 - (-4)| = 8\). Using the distance formula breaks these concepts into manageable steps, helping to visualize the dimensions of geometric figures more clearly.

Standard Form of an Ellipse

An ellipse in mathematics is represented in a specific form that highlights its key features—the standard form of an ellipse. This format allows for easy identification of the ellipse's major and minor axes.

• The standard form of an ellipse with a center at the origin \((0,0)\) is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\),

where \(a\) and \(b\) represent the semi-major and semi-minor axes, respectively. The orientation depends on the values of \(a\) and \(b\).

• If \(a > b\), the ellipse is stretched along the x-axis.
• If \(b > a\), it stretches along the y-axis, as in our given example equation.

By rewriting the original equation \(4x^2 + y^2 = 16\) into its standard form \(\frac{x^2}{4} + \frac{y^2}{16} = 1\), we clearly see it stretches along the y-axis. This form simplifies comparing and understanding ellipses, essential for geometric analysis.