Problem 71 - Intermediate Algebra

Question

What happens to the graph of the equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) when \(a=b ?\)

Step-by-Step Solution

Answer

The graph becomes a circle with radius \(a\).

Step 1: Identify the Type of Equation

The given equation \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) is the standard form for the equation of an ellipse. This ellipse is centered at the origin \((0, 0)\) with semi-major axis \(a\) along the x-axis and semi-minor axis \(b\) along the y-axis.

Step 2: Set Condition for Special Case

The problem specifies the condition \(a = b\). Substitute this condition into the equation, which transforms it into \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}=1\).

Step 3: Simplify the Equation

Factor out \(\frac{1}{a^{2}}\) from both terms, leading to \(\frac{x^{2} + y^{2}}{a^{2}} = 1\). Multiply through by \(a^{2}\) to get \(x^{2} + y^{2} = a^{2}\).

Step 4: Determine the Shape

The equation \(x^{2} + y^{2} = a^{2}\) is the equation for a circle centered at the origin \((0, 0)\) with radius \(a\).

Step 5: Conclusion

When \(a = b\), the graph of the equation changes from an ellipse to a circle with radius \(a\).

Key Concepts

Standard Form of an Ellipse

Ellipses are fundamental shapes in geometry, and one of their most significant features is their equation. In its standard form, an ellipse is described by the equation \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \). This equation indicates that the ellipse is centered at the origin \((0, 0)\). The terms \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes, respectively.

• The semi-major axis is the longer radius and is aligned with the x-axis if \(a > b\).
• The semi-minor axis is the shorter radius and is aligned with the y-axis if \(b > a\).

The lengths of these axes determine the shape's elongated appearance. By manipulating \(a\) and \(b\), one can change the proportions of the ellipse, stretching it along the x-axis or y-axis accordingly. It is essential to understand these parts to determine how repositioning or reshaping occurs under different mathematical conditions.

Special Cases in Conic Sections

Conic sections include circles, ellipses, parabolas, and hyperbolas. Each is defined by specific conditions within their equations. A special case of an ellipse occurs when the semi-major and semi-minor axes are equal, that is when \(a = b\). Under this condition, the ellipse takes the shape of a circle.

Substituting \(a = b\) into the standard form of an ellipse equation \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{a^{2}} = 1\), the equation simplifies to \(\frac{x^{2} + y^{2}}{a^{2}} = 1\). When multiplied through by \(a^{2}\), it becomes \(x^{2} + y^{2} = a^{2}\). This is the standard equation for a circle with radius \(a\) centered at the origin.

• Circle: unique case of ellipse where axes' lengths are equal.
• Equation simplifies to represent a circle of radius \(a\) on both axes.

Equation Transformation

Equation transformations are powerful tools that allow us to understand different geometric shapes. By transforming the equation of an ellipse where \(a\) and \(b\) are equal, we convert it into the equation of a circle. This transformation illustrates how equations can represent different shapes under specific conditions.

The process of simplifying or manipulating the equation to highlight different forms is key to exploring the relationships between various conic sections. In our case, setting \(a = b\) converts the equation \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) to \(x^{2} + y^{2} = a^{2}\), showing a clear geometric shift from an ellipse to a circle.

• Transformation demonstrates the flexibility of mathematical equations.
• Abstraction highlights geometric changes and properties.