Skip to main content
Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 41
Chapter 3, Problem 41

Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. x2+y2+4x+14y=-54

Verified step by step guidance
1
Start by rewriting the given equation: \(x^2 + y^2 + 4x + 14y = -54\).
Group the \(x\) terms and \(y\) terms together: \((x^2 + 4x) + (y^2 + 14y) = -54\).
Complete the square for the \(x\) terms: take half of 4, which is 2, and square it to get 4. Add and subtract 4 inside the equation.
Complete the square for the \(y\) terms: take half of 14, which is 7, and square it to get 49. Add and subtract 49 inside the equation.
Rewrite the equation as perfect square trinomials: \((x + 2)^2 - 4 + (y + 7)^2 - 49 = -54\). Then, combine constants on the right side to find the standard form of the circle equation.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Form of a Circle Equation

The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Converting a general quadratic equation into this form helps identify the circle's properties.
Recommended video:
5:18
Circles in Standard Form

Completing the Square

Completing the square is a method used to rewrite quadratic expressions as perfect square trinomials. This technique is essential to transform the given equation into the standard form of a circle.
Recommended video:
06:24
Solving Quadratic Equations by Completing the Square

Interpreting the Radius and Existence of the Circle

After rewriting the equation, the radius squared (r^2) must be positive for a circle to exist. If r^2 = 0, the graph is a single point; if r^2 is negative, the graph does not exist in the real plane.
Recommended video:
5:18
Circles in Standard Form
Related Practice
Textbook Question

Find the slope of the line satisfying the given conditions. through (2, -1) and (-3, -3)

Textbook Question

Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b).

midpoint (6a,6b)(6a, 6b), endpoint (3a,5b)(3a, 5b)

Textbook Question

Graph each equation. ƒ(x) = x

1
views
Textbook Question

Give a rule for each piecewise-defined function. Also give the domain and range.

Textbook Question

Plot each point, and then plot the points that are symmetric to the given point with respect to the (a) x-axis, (b) y-axis, and (c) origin. (-4, -2)

Textbook Question

Find the slope and y-intercept of each line, and graph it. x+2y = -4