Problem 75 - Contemporary Precalculus

Question

Consider the ellipse whose equation is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 .\) Show that if \(a=b,\) then the graph is actually a circle.

Step-by-Step Solution

Answer

Question: Show that if a=b in the ellipse equation, the graph is a circle. Answer: If a=b in the ellipse equation, it simplifies to the circle equation (x-0)^2+(y-0)^2=b^2 with center (0,0) and radius b.

Step 1: Identify the ellipse equation

We are given the ellipse equation in the form \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 .\)

Step 2: Replace a with b

Since we are given that \(a = b\), we can replace \(a\) with \(b\) in the equation: \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{b^{2}}=1 .\)

Step 3: Simplify the equation

Now, we can factor out the common term \(\frac{1}{b^2}\) from both terms in the equation: \(\frac{1}{b^{2}}(x^{2}+y^{2})=1.\)

Step 4: Rearrange the equation

To find the equation of the circle, we need to get it in the form \((x-h)^2+(y-k)^2=r^2\). We can start by multiplying both sides of the equation by \(b^2\): \(x^2+y^2=b^2.\)

Step 5: Identify the circle equation

Now, we can see that the equation is in the form of a circle with center \((h, k) = (0, 0)\) and radius \(r = b\). The equation is \((x-0)^2+(y-0)^2=b^2\). Therefore, if \(a = b\), the graph of the ellipse is actually a circle.

Key Concepts

Ellipse Equation

The equation of an ellipse is a fundamental concept in the study of conic sections. In its standard form, the equation is given as \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}=1\). This expression represents an ellipse centered at the origin (0,0) of the coordinate plane. The parameters \(a\) and \(b\) are referred to as the semi-major and semi-minor axes of the ellipse, respectively.

When \(a > b\), the ellipse is elongated along the x-axis, and when \(a < b\), it is elongated along the y-axis. Understanding this equation is crucial because it encapsulates all elliptical shapes.

Importance of the Semi-Major and Semi-Minor Axes

The values of \(a\) and \(b\) not only determine the shape of the ellipse but also its flatness or roundness. If \(a = b\), the ellipse becomes a perfect circle, as both axes are equal, leading to the same curvature in all directions.

Transformation into a Circle

This transformation can be visualized by substituting \(a\) with \(b\), simplifying the equation to \(x^2+y^2=b^2\), which matches the standard form of a circle's equation with a radius of \(b\). Recognizing the relationship between the values of \(a\) and \(b\) is beneficial for graphing ellipses and understanding their geometric properties.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double napped cone. They include ellipses, circles, parabolas, and hyperbolas. Each of these shapes has unique properties and equations, but they are all related through the angle of intersection and the relative position of the plane to the cone's axis.

Ellipses and circles are both types of conic sections that result from a plane intersecting the cone at an angle to the base. If this angle results in a closed curve that is not a perfect circle, then the shape is an ellipse. Circles are a special case of ellipses where the plane is perpendicular to the cone's axis, leading to an equal distance from the center to any point on the curve.

The Circle as a Special Ellipse

It is fundamental to understand that a circle can be thought of as an ellipse with equal axes, emphasizing how these shapes are connected within the family of conic sections. Learning these relationships deepens comprehension of the geometric principles governing conic sections and assists in generalizing the concepts, which is especially valuable when working with abstract equations.

Circle Properties

A circle is one of the most fundamental and symmetric shapes in geometry with a set of unique properties. Its defining equation in the Cartesian coordinate system is \( (x-h)^2 + (y-k)^2 = r^2 \), where \( (h, k) \) is the center of the circle, and \( r \) is the radius. This equation tells us that a circle consists of all points in a plane that are at a constant distance, the radius, from a fixed point, the center.

Circles are conic sections that can be derived from the equation of an ellipse when the semi-major and semi-minor axes are equivalent. The formula simplifies into the circle equation, highlighting the circle's perfect symmetry around its center.

Visualizing Circle Symmetry

Circle symmetry is evident in every diameter passing through its center, dividing it into two equal halves. This inherent symmetry is what makes circles a unique member of the conic sections family, and understanding their properties is key to grasping the nature of these geometric shapes. Recognizing the circle's distinctive characteristics aids in solving various geometric problems and analyzing the behavior of points around a central location.