Problem 41 - 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. \(9x^2+4y^2+36x-24y+36=0\)

Step-by-Step Solution

Answer

The given equation forms a circle with center at (-2,3), radius 1, vertices at (-3,3) and (-1,3), foci at (-2,3) and eccentricity of 0.

Step 1: Rewrite in standard form

Group the x and y terms and complete the square to rewrite the equation in standard form: Dividing entire equation by 4 would give us \(2.25x^2 + 4x + 9y^2 - 6y +9 = 0\)which can be rewritten as: \((x + 2)^2 + 1y^2 - 6y +9 = 0\) which can be further simplified to:\((x + 2)^2 + (y - 3)^2 = 1 \)

Step 2: Identify the conic and find the center

The equation \((x + 2)^2 + (y - 3)^2 = 1 \) is in the standard form of a circle, where the center is given by (-2,3). The coefficient of x and y are both 1, indicating that this conic section is indeed a circle.

Step 3: Radius, Vertices and Foci

For a circle, the radius, \(r\), is given by \(\sqrt{h}\) where \(h\) is the constant on the right side of the equation.Thus, radius, r = \(\sqrt{1} = 1\)Vertices of a circle are endpoints of the diameter. Therefore our vertices can be found at (-2-1,3) and (-2+1,3) or (-3,3) and (-1,3)For a circle, the foci coincide with the center, thus foci is at (-2,3)

Step 4: Eccentricity of the conic

Since it's a circle, the eccentricity is 0 as the two foci coincide.

Step 5: Sketch the graph

Plot the center, vertices, and foci on a graph. Draw a circle with a radius of 1. This gives us a sketch of the conic.

Key Concepts

Circle

A circle is one of the simplest forms of conic sections. It is defined as the set of all points in a plane that are equidistant from a fixed point, known as the center. The standard equation for a circle is \[ (x - h)^2 + (y - k)^2 = r^2 \] where

• \((h, k)\) is the center of the circle
• \(r\) is the radius

The equation given, \((x + 2)^2 + (y - 3)^2 = 1\), shows a circle with a center at \((-2, 3)\) and a radius of 1. This indicates that every point on the circle is 1 unit away from the center. Identifying a circle from its equation helps us understand its geometric properties easily.
Circles are unique among conic sections due to their perfect symmetry and constant radius.

Ellipse

An ellipse resembles a stretched circle and can be described as the set of all points for which the sum of the distances from two fixed points (foci) is constant. Though similar in appearance to a circle, an ellipse has different lengths for major and minor axes. The standard form of the ellipse’s equation is \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \] where:

• \((h, k)\) is the center
• \(a\) is the semi-major axis
• \(b\) is the semi-minor axis

Ellipses become important when exploring planetary orbits and satellite paths. In this exercise, we identified the conic section as a circle rather than an ellipse since the coefficients of \(x^2\) and \(y^2\) are equal, resulting in a uniformly round shape rather than an elongated form typical of ellipses.

Eccentricity

Eccentricity helps us determine how much a conic section deviates from being circular. It is represented by the symbol \(e\). For different conic sections, the eccentricity varies:

• Circle: \(e = 0\)
• Ellipse: \(0 < e < 1\)
• Parabola: \(e = 1\)
• Hyperbola: \(e > 1\)

Since a circle has an eccentricity of 0, it means there's no deviation from being circular, a perfect round shape. The eccentricity is a crucial concept in understanding how different conic sections relate to each other. In ellipses, it aids in understanding the shape's elongation and how close it is to resembling a circle.

Vertices

Vertices in conic sections are key points that help in defining the shape and extent of the figure. For a circle, vertices are typically the endpoints of the diameter. In our circle, with a center at \((-2, 3)\) and a radius of 1, the vertices are at the points \((-3, 3)\) and \((-1, 3)\). These points represent the farthest edges along a straight line passing through the center.

• For a circle: Defined by its diameter.
• For an ellipse: Located at either end of the major axis.

Understanding vertices helps in graphing and conceptualizing the size and position of conic sections in a coordinate plane.

Foci

Foci are specific points in conic sections from which distances help define the shape's properties.

• In a circle, the foci coincide with the center, making a circle unique as it involves only one central point, \((-2, 3)\) in this exercise.
• In an ellipse, the foci are two distinct points located along the major axis.

The role of foci differs among conic sections. For circles, the foci overlap with the center, reinforcing the circle’s symmetry and uniform radius. In ellipses, the distance to each focus varies, emphasizing their elongated shape. Recognizing foci locations helps clarify the geometric structure of different conic sections.