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Chapter 1: Problem 37

Determine the center and radius of each circle and sketch the graph. $$y^{2}=25-(x+1)^{2}$$

Short Answer

Expert verified
Center: (-1, 0), Radius: 5

Step by step solution

01

Rewrite the Equation

Start by rewriting the given equation in a standard form. The equation provided is: y^{2} = 25 - (x+1)^{2}To rewrite it, express it in the standard form of a circle, which is otes: (x-h)^{2} + (y-k)^{2} = r^{2}(x-h)^{2} + (y-k)^{2} = r^{2}
02

Complete the square for x-term

To fit the standard form, expand the right side of the equation:y^{2} = 25 - [(x+1)^{2}]y^{2} = 25 - [x^{2} + 2x + 1] y^{2} = 25 - x^{2} - 2x - 1y^{2} = - x^{2} - 2x - 24

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Key Concepts

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

Standard Form of a Circle
In precalculus, understanding the standard form of a circle's equation is crucial. The standard form is \((x - h)^2 + (y - k)^2 = r^2\). Here, \((h, k)\) represents the circle's center and \((r)\) is the radius. This format is beneficial because it directly indicates the circle's center and radius, making it easier to graph.
Completing the Square
To put an equation into the standard form, completing the square is often a key step. This technique helps in rewriting quadratic expressions by converting them into perfect square trinomials. For example, if given \(y^2 = 25 - (x + 1)^2\), we start by expanding and simplifying the equation:
  • Expand the right side: \(y^2 = 25 - (x^2 + 2x + 1)\)
  • Simplify: \(y^2 = 24 - x^2 - 2x\)
This technique transforms the equation into a recognizable circle format, aiding in identifying the center and radius.
Graphing Circles
Once the equation is in standard form, graphing becomes straightforward. Here鈥檚 how to sketch it:
  • Identify the center \((h, k)\).
  • Calculate the radius \((r)\) from the equation \((r^2)\).
For instance, with our equation \(y^2 = 25 - (x + 1)^2\), completing the square and reformatting lets us find:
  • Center: \((h, k) = (0, 0)\)
  • Radius: \(r = 5\)
Plot the center first, then use the radius to draw the circle accurately.

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Most popular questions from this chapter

Write an inequality of the form \(|x-a| < k\) or of the form \(|x-a| > k\) so that the inequality has the given solution set. HINT: \(|x-a| < k\) means that \(x\) is less than \(k\) units from \(a\) and \(|x-a|>k\) means that \(x\) is more than \(k\) units from \(a\) on the number line. $$(-\infty,-1) \cup(5, \infty)$$

Solve each equation. $$x+0.1 x=121$$

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Find an exact solution to each problem. If the solution is irrational, then find an approximate solution also. Find the radius of the circle \(x^{2}+y^{2}=3 y\)
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