Problem 48 - 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. \(x^2+y^2-4x+6y-3=0\)

Step-by-Step Solution

Answer

The given equation represents a circle with center at (2,-3), radius 4, and vertices at (6,-3),(-2,-3),(2,1) and (2,-7). The foci is (2,-3) and the eccentricity is 0.

Step 1: Rearrange and Complete the Square

First arrange the equation and complete the square for both x and y terms. The given equation is \(x^2+y^2-4x+6y-3=0\). Rearranging and completing the square: \((x^2-4x) + (y^2+6y) = 3 \), \((x^2-4x+4) + (y^2 + 6y + 9) = 3 + 4 + 9\), \((x-2)^2 + (y+3)^2 = 16 \) . This is the standard form of a circle.

Step 2: Identify the Conic and its Properties

The standard form of a circle equation is \((x-h)^2 + (y-k)^2 = r^2 \). By comparing this general equation of a circle with the equation \((x-2)^2 + (y+3)^2 = 16\), we can derive that the center of the circle (h,k) is (2,-3), and the radius r is \( \sqrt{16} \) = 4

Step 3: More Properties

For a circle, the vertices are the points on the circle that are aligned with the center on the x and y axis. It implies that for this circle there are 4 vertices which are (2+4, -3) = (6,-3), (2-4,-3) = (-2,-3), (2,-3+4) = (2,1) and (2,-3-4) = (2,-7). The foci of a circle coincide with the center, so the foci is (2,-3). The eccentricity of a circle is 0.

Step 4: Sketch the Graph

To sketch the graph, draw a circle with center at (2,-3), radius 4, and vertices at (6,-3),(-2,-3),(2,1) and (2,-7).

Key Concepts

Circle

In geometry, a circle is a simple yet fascinating shape. It's defined as the set of all points in a plane that are equidistant from a fixed point called the center. This fixed distance from the center to any point on the circle is the radius. Circles are unique because they have a round shape and are perfectly symmetrical. When dealing with equations, the standard form of a circle is

• \[(x-h)^2 + (y-k)^2 = r^2\]

Here,

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

Circles have many real-world applications, from wheels to watches, and play a critical role in mathematics and engineering.

Radius

The radius of a circle is a crucial measurement. It is the distance from the center of the circle to any point on its boundary. In our equation example, the circle's equation is

• \[(x-2)^2 + (y+3)^2 = 16\]

To find the radius, we take the square root of 16, which gives us a radius of 4. The radius is essential not only in defining the size of the circle but also in calculating other properties like the area and circumference. Remember, the uniform distance ensures that all points lie on the circle.

Center of Circle

The center of a circle is the point from which all points on the circle are equally distant. It serves as the circle’s midpoint. When we look at the circle equation

• \[(x-h)^2 + (y-k)^2 = r^2\]

we identify

• the center as \((h, k)\).

In our case, the equation

• \[(x-2)^2 + (y+3)^2 = 16\]

reveals the center is located at \((2, -3)\). The center is foundational in understanding the circle's symmetry, and it often serves as a reference point for graphing.

Vertices of Circle

In the context of circles, vertices refer to significant points along the circle's path. These are usually the points that lie directly above, below, or to either side of the center, aligned along the main axes. For example, in the problem we explored, the circle has vertices at coordinates:

• \((6, -3)\),
• \((-2, -3)\),
• \((2, 1)\),
• \((2, -7)\).

These points result from moving the radius length along the x and y axes starting from the center at \((2, -3)\). Vertices help in visualizing and sketching the circle with precision.

Eccentricity

Eccentricity is a measure used to determine how much a conic section deviates from being circular. It's an essential concept in conic sections like ellipses and hyperbolas. However, for a circle, the eccentricity is always zero. This is because all the points are uniformly distant from the center, indicating no deviation. The formula for eccentricity in general conics is:

• \[e = \frac{c}{a}\]

where

• \(c\) is the distance between the foci and the center, and
• \(a\) is the distance from the center to a vertex.

But in a circle, since the foci coincide with the center, \(c = 0\), making the eccentricity \(e = 0\). This beautiful property tells us that circles are perfectly round and is a reason why they’re unique among conic sections.