Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 13: Problem 55
Find the center and the radius of each circle. Then graph the circle. $$ x^{2}+y^{2}+10 y-75=0 $$
Short Answer
Expert verified
Center: (0, -5), Radius: 10
Step by step solution
01
- Rewrite the equation
Start by rewriting the given equation in a more recognizable form for a circle. The given equation is: \[ x^{2}+y^{2}+10y-75=0 \]Separate the terms involving y:\[ x^{2} + (y^{2} + 10y) - 75 = 0 \]
02
- Complete the square for y
To complete the square for the terms involving y, follow these steps:1. Take the coefficient of y (which is 10), divide it by 2, and then square it: \[ \frac{10}{2} = 5 \text{ and } 5^2 = 25 \]2. Add and subtract this square within the equation:\[ x^{2} + (y^{2} + 10y + 25 - 25) - 75 = 0 \]3. Simplify within the parentheses to form a perfect square trinomial:\[ x^{2} + (y + 5)^2 - 100 = 0 \]
03
- Convert to standard form
Move the constant term to the other side of the equation to rewrite it in the standard form of a circle鈥檚 equation: \[ x^{2} + (y + 5)^{2} = 100 \]
04
- Identify the center and radius
Now, identify the center and the radius of the circle from the standard form:1. The standard form of a circle鈥檚 equation is:\[ (x - h)^{2} + (y - k)^{2} = r^2 \]2. Here, the given equation is:\[ x^{2} + (y + 5)^{2} = 100 \]3. Thus, comparing the two, the center (h, k) is at \[ (0, -5) \] and the radius r is:\[ r = \sqrt{100} = 10 \]
05
- Graph the circle
Graph the circle with the following details:1. Plot the center of the circle at (0, -5).2. Draw a circle around this center with a radius of 10 units.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a method used to convert a quadratic equation into a more manageable form. This technique is particularly helpful when dealing with equations of circles.
Let's break down the steps:
Understanding this method is essential for converting and solving equations of circles and other quadratic equations efficiently.
Let's break down the steps:
- First, identify the quadratic term you need to form a complete square. In this exercise, we looked at the terms involving \(y\): \(y^{2}+10y\).
- Next, take the coefficient of the linear term, which is 10, divide it by 2 to get 5, and then square it to get 25.
- Add and subtract this squared value inside the equation: \(x^{2}+(y^{2}+10y+25-25)-75=0\).
- Now, \(y^{2}+10y+25\) forms a perfect square trinomial, which can be written as \((y+5)^2\).
Understanding this method is essential for converting and solving equations of circles and other quadratic equations efficiently.
Standard Form of Circle
The standard form of a circle's equation is \((x-h)^2+(y-k)^2=r^2\).
This form is very useful because it directly reveals the circle's center and radius in a clear, concise way.
The general structure follows these principles:
Comparing this to \((x-h)^{2}+(y-k)^{2}=r^{2}\):
This form is very useful because it directly reveals the circle's center and radius in a clear, concise way.
The general structure follows these principles:
- \(h\) and \(k\) represent the coordinates of the center of the circle.
- \(r\) is the radius of the circle, but in the equation, it appears as \(r^2\).
Comparing this to \((x-h)^{2}+(y-k)^{2}=r^{2}\):
- We see that \(h=0\) and \(k=-5\), making the center \( (0, -5) \).
- The radius \(r\) is \(\root100=10\).
Graphing Circles
Graphing a circle involves plotting its center and then drawing the curve by maintaining a constant distance (the radius) from this center point in all directions.
Here's how you can graph a circle step-by-step:
This skill often comes in handy in coordinate geometry and various real-world applications like planning and design.
Here's how you can graph a circle step-by-step:
- First, identify the center of the circle from the standard form of the equation. In our example, the center is at \( (0, -5) \).
- Next, determine the radius. For our circle, the radius is 10 units.
- Plot the center point on your graph. Here, you'll mark the point at \((0, -5) \).
- From the center, measure 10 units in all directions to outline the circle. This includes upward, downward, right, and left from the center.
- Finally, connect these points in a smooth, round shape to complete the circle.
This skill often comes in handy in coordinate geometry and various real-world applications like planning and design.
