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Chapter 10: Problem 62

Sketch the graph of each equation. If the graph is a parabola, find its vertex. If the graph is a circle, find its center and radius. $$x^{2}+(y+5)^{2}=5$$

Short Answer

Expert verified
The center of the circle is (0,-5) and the radius is \(\sqrt{5}\).

Step by step solution

01

Identify the Equation Type

The equation given is \[ x^{2} + (y+5)^{2} = 5 \]This is the standard form of a circle's equation: \[ (x - h)^2 + (y - k)^2 = r^2 \] Comparing, we see this equation represents a circle.
02

Determine the Center

Using the circle equation \( (x - h)^2 + (y - k)^2 = r^2 \), we identify \( h = 0 \) and \( k = -5 \). Thus, the center of the circle is at the point \((0, -5)\).
03

Determine the Radius

In the circle's equation, the value on the right side \( r^2 = 5 \). Therefore, the radius is \[ r = \sqrt{5} \].
04

Sketch the Graph

Plot the center of the circle at \((0, -5)\) on the coordinate plane. Using the radius \(\sqrt{5}\), draw the circle around the center. Ensure the circle has a radius of approximately \(2.24\), since \(\sqrt{5} \approx 2.24\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form Equation
Understanding that the equation \(x^{2}+(y+5)^{2}=5\) fits the mold of a standard circle equation is crucial. The standard form of a circle's equation is expressed as \((x - h)^2 + (y - k)^2 = r^2\).
Here, \(h\) and \(k\) represent the x and y coordinates of the circle's center, and \(r\) is the radius.
By comparing equations, we can identify the values of \(h\), \(k\), and \(r^2\) directly from the equation. With this form, understanding other parts of the circle becomes simple.
Center of a Circle
To find the center of the circle from its equation \((x - h)^2 + (y - k)^2 = r^2\), you need to extract the values of \(h\) and \(k\).
These values are the coordinates of the circle's center. In this problem:
  • The term \((x - h)^2\) suggests \(h = 0\), since \(x\) is not adjusted by any number.

  • The term \((y - k)^2\) corresponds to \((y + 5)^2\), indicating that \(k = -5\).
Therefore, the circle's center is located at the point \((0, -5)\) on the coordinate plane.
Radius Calculation
Finding the radius of a circle involves recognizing the format of the equation \(r^2\).
In the equation \(x^{2}+(y+5)^{2}=5\), the number \(5\) is equivalent to \(r^2\).
For the radius \(r\), take the square root of \(5\), which results in \(r = \sqrt{5}\).
This evaluates to approximately \(2.24\). It's essential to perform this calculation since the radius is the distance from the center to any point on the circle.
Graph Sketching
Sketching a graph of a circle involves a few clear steps to visualize the mathematical properties.
Start by plotting the center point at \((0, -5)\) on a coordinate plane. This is the fixed point from which the circle extends outward.
  • With the center plotted, use the radius \(\sqrt{5}\) (approximately \(2.24\)) to draw the circle.

  • Measuring from the center, mark several points exactly \(2.24\) units away.

  • Ensure that these points create a smooth round shape as you connect them back to the starting point.
Following these steps will help accurately sketch the circle's graph, giving a clear view of its boundary.

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Most popular questions from this chapter

Solve each nonlinear system of equations for real solutions. $$ \left\\{\begin{array}{l} {y=x^{2}+2} \\ {y=-x^{2}+4} \end{array}\right. $$

Graph each inequality in two variables. $$ 3 x-y \leq 4 $$

Solve each nonlinear system of equations for real solutions. $$ \left\\{\begin{array}{l} {x^{2}+2 y^{2}=2} \\ {x-y=2} \end{array}\right. $$

Solve each nonlinear system of equations for real solutions. $$ \left\\{\begin{aligned} x^{2}+y^{2} &=25 \\ 3 x+4 y &=0 \end{aligned}\right. $$

If you are given a list of equations of circles, parabolas, ellipses, and hyperbolas, explain how you could distinguish the different conic sections from their equations.
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