Problem 31 - Precalculus with Limits

Question

In Exercises 29-52, identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(\dfrac{x^2}{25}+\dfrac{y^2}{25}=1\)

Step-by-Step Solution

Answer

This is a circle with a center at (0,0), and a radius of 5. The concepts of vertices, foci, and eccentricity aren't applicable to a circle.

Step 1: Identify the Conic

The given equation is \(\frac{x^2}{25} + \frac{y^2}{25} = 1\). This is the standard equation of a circle or an ellipse. The coefficients of \(x^2\) and \(y^2\) are the same: \(1/25\). If the coefficients are equal, the conic is a circle.

Step 2: Find the Center

The center of the circle is always at the point (h, k) where h and k are the values that would make the equations in the form \((x-h)^2\) or \((y-k)^2\). Since there are no such terms in our equation, the center of the circle is at the origin, (0,0).

Step 3: Find the Radius

For the equation in the form \(\frac{x^2}{r^2}+\frac{y^2}{r^2}=1\), the radius of the circle is given by \(r = \sqrt{25}\), so the radius is 5.

Step 4: Determine Vertices, Foci, and Eccentricity

For a circle, the concepts of vertices, foci, and eccentricity aren't applicable.

Step 5: Sketch the Graph

The circle can be graphed on a coordinate plane. Plot the center at (0,0). From the center, count 5 units up, down, left, and right. Connect those points to graph the circle.

Key Concepts

Circle Equations

Understanding the equation of a circle is fundamental when identifying conic sections. A circle equation in the standard form is represented by \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center of the circle, and \( r \) is the radius. If an equation resembles \( x^2 + y^2 = r^2 \), with both \( x^2 \) and \( y^2 \) having the same coefficient and without any \( h \) and \( k \) values, it indicates that the circle's center is at the origin \( (0,0) \) and its radius is the square root of the constant term.

For example, \( \dfrac{x^2}{25} + \dfrac{y^2}{25} = 1 \) represents a circle centered at the origin with a radius of 5, because rewriting it as \( x^2 + y^2 = 25 \) matches the standard circle form with \( r = \sqrt{25} \), which equals 5.

Ellipse Equations

An ellipse equation generally looks like \( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \) where \( (h, k) \) denotes the center, and \( a \) and \( b \) are the distances from the center to the vertices along the major and minor axes, respectively. Unlike a circle, the coefficients of \( x^2 \) and \( y^2 \) are different, signifying the varied lengths of the axes.

For ellipses, vertices are located at \( (h \pm a, k) \) and \( (h, k \pm b) \) along the major and minor axes. The foci are points inside the ellipse along the major axis, found using \( c^2 = a^2 - b^2 \) where \( c \) is the distance from the center to each focus. The eccentricity, given by \( e = \frac{c}{a} \) for ellipses with a horizontal major axis or \( e = \frac{c}{b} \) for a vertical major axis, measures the elongation of the ellipse.

Conic Sections

Conic sections arise from the intersection of a plane with a double-napped cone. The four basic types of conic sections are circles, ellipses, parabolas, and hyperbolas. Each can be defined algebraically by a general second-degree equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \).

• A circle occurs when \( A = C \) and \( B = 0 \) with \( A > 0 \).
• An ellipse exists if \( A eq C \) and \( B = 0 \) with both \( A \) and \( C \) being positive.
• A parabola forms when either \( A = 0 \) or \( C = 0 \) but not both.
• A hyperbola is characterized by \( A eq C \) and often \( B eq 0 \) with one of \( A \) or \( C \) being negative.

By determining the values of \( A \) and \( C \) and whether \( B \) is zero or not, one can discern the type of conic section represented by a given equation.

Graphing Conics

Graphing conics involves plotting their shapes on a coordinate grid based on their defining equations. For circles and ellipses, this means determining the center, radius (for circles) or axes lengths (for ellipses), and direction of the axes.

Steps for Graphing Circles

• Locate the center \( (h, k) \).
• Plot points at a distance \( r \) from the center in all directions.
• Draw a smooth curve connecting these points to form a circle.

Steps for Graphing Ellipses

• Plot the center \( (h, k) \).
• Mark the vertices along the major and minor axes.
• Identify the foci if necessary.
• Connect these points with a smooth, oval shape.

For the given exercise, plotting the origin as the center and moving 5 units in each direction before drawing the circle satisfies the requirement for graphing the given conic.