Problem 60 - Intermediate Algebra

Question

We know that \(x^{2}+y^{2}=25\) is the equation of a circle. Rewrite the equation so that the right side is equal to \(1 .\) Which type of conic section does this equation form resemble? In fact, the circle is a special case of this type of conic section. Describe the conditions under which this type of conic section is a circle.

Step-by-Step Solution

Answer

The rewritten equation is \(\frac{x^2}{25} + \frac{y^2}{25} = 1\). It resembles an ellipse, which becomes a circle when \(a = b\).

Step 1: Understand the Given Equation

The given equation is \(x^2 + y^2 = 25\), which represents a circle with center at the origin (0,0) and a radius of 5.

Step 2: Change the Right Side to 1

In order to transform the equation so that the right side becomes 1, we need to divide all terms of the equation by 25. This gives us \(\frac{x^2}{25} + \frac{y^2}{25} = 1\).

Step 3: Identify the Conic Section

The equation \(\frac{x^2}{25} + \frac{y^2}{25} = 1\) resembles the standard form of an ellipse, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where both \(a\) and \(b\) equal 5 in this case.

Step 4: Analyze the Special Case for a Circle

For an ellipse to be a circle, the lengths of its semi-major and semi-minor axes must be equal. In the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), this means \(a\) must equal \(b\). In \(\frac{x^2}{25} + \frac{y^2}{25} = 1\), both denominators are 25, meaning \(a = b = 5\). Hence, it is a circle.

Key Concepts

Equation of a Circle

The equation of a circle in its simplest form is written as \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. This equation represents a circle centered at the origin, point (0, 0). To find the radius, you simply take the square root of the number on the right side of the equation. For instance, in the equation \(x^2 + y^2 = 25\), the radius \(r\) is 5 because \(\sqrt{25} = 5\).
Circle equations help you easily identify the size and location of the circle on a graph. They are the starting point for understanding how circles can be represented in coordinate geometry.

Ellipse

An ellipse is a type of conic section that looks like an elongated circle. The basic equation of an ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a\) and \(b\) represent the semi-major and semi-minor axes of the ellipse.

• When \(a > b\), the ellipse is elongated along the x-axis.
• Conversely, when \(b > a\), it stretches along the y-axis.

Ellipses have two focal points, and the sum of the distances from any point on the ellipse to these foci remains constant. This property is what makes ellipses unique compared to other conic sections.
The shape of ellipses is widely seen in planetary orbits, making them an important part of studies in astronomy and physics.

Standard Form of Ellipse

The standard form of an ellipse's equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) allows us to identify the ellipse's orientation and dimensions. Every ellipse has axes of symmetry, and this form helps easily identify the lengths of these axes. The axes are measured as:

• Length of the semi-major axis: \(a\)
• Length of the semi-minor axis: \(b\)

By comparing \(a\) and \(b\), we can quickly understand the direction in which the ellipse is elongated. This form is particularly helpful because it simplifies calculations while transforming ellipses in geometric space. A special property of this form is that when \(a = b\), the ellipse becomes a circle, marking a key point when learning about the connection between circles and ellipses.

Special Case of Conic Sections

Conic sections encompass many familiar shapes, such as circles, ellipses, parabolas, and hyperbolas. A circle is actually a special case of an ellipse, specifically when the ellipse's axes are equal, meaning \(a = b\).
This situation leads to both equations \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) and \(x^2 + y^2 = r^2\) being equivalent. It shifts our perspective from a broad category into something specific. Conic sections in general are based on cutting a cone in various ways, but when we hone in on equal axes in an ellipse, we see the creation of a perfect circle.
Understanding this special case is crucial because it bridges concepts, showing that seemingly different mathematical objects might be closely related.