Write the Standard Form of the Equation for this Circle

AI Thread Summary
To find the standard form of the equation of a circle given the endpoints of a diameter, first calculate the midpoint of the endpoints to determine the center (h, k). The midpoint formula is applied to the points (11, -5) and (3, 15), yielding the center at (7, 5). Next, use the distance formula to find the radius by calculating the distance from the center to one of the endpoints. Finally, plug the values into the standard equation (x - h)² + (y - k)² = r², resulting in (x - 7)² + (y - 5)² = 116. The problem is resolved with the correct equation derived from these steps.
nycmathguy
Homework Statement
Write standard form of the equation of a circle.
Relevant Equations
Equation of a circle not centered at the origin.

(x - h)^2 + (y - k)^2 = r^2
Chapter 1, Section 1.2

Write the standard form of the equation of the circle with the given characteristics.

74. Endpoints of a diameter: (11, −5), (3, 15)

I want to know if the following steps are correct for me to answer the above question.

Steps:

1. Find the distance between the points.

2. Divide the distance by 2 to find the radius.

3. Plug into (x - h)^2 + (y - k)^2 = r^2

You say?
 
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nycmathguy said:
Homework Statement:: Write standard form of the equation of a circle.
Relevant Equations:: Equation of a circle not centered at the origin.

(x - h)^2 + (y - k)^2 = r^2

Steps:

1. Find the distance between the points.

2. Divide the distance by 2 to find the radius.

3. Plug into (x - h)^2 + (y - k)^2 = r^2
Looks reasonable so far, but how do you use those two endpoints to find h and k? :smile:
 
Do you know the midpoint formula?
 
cbarker1 said:
Do you know the midpoint formula?

Yes, I know the midpoint formula. What about it?
 
berkeman said:
Looks reasonable so far, but how do you use those two endpoints to find h and k? :smile:
I would plug the x and y values of the endpoints in the standard form of an equation of a circle to find h and k. Yes?
 
nycmathguy said:
I would plug the x and y values of the endpoints in the standard form of an equation of a circle to find h and k. Yes?
Sorry, I have no idea what that means. Instead, I would follow this hint:
cbarker1 said:
Do you know the midpoint formula?
 
nycmathguy said:
You say?
I say that once again, you are not following through.

Draw the two points on a graph. Calculate the midpoint numerically and see if it looks right on the graph. When you have the midpoint, write the equation of the circle. There's more for a full follow-through.
 
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nycmathguy said:
Yes, I know the midpoint formula. What about it?
How is the mid-point of a diameter of a circle related to the location of the circle's center ?
 
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nycmathguy said:
Yes, I know the midpoint formula. What about it?
So what is the formula, then?
 
  • #10
SammyS said:
How is the mid-point of a diameter of a circle related to the location of the circle's center ?

The midpoint of a circle divides the diameter into two radii.
 
  • #11
nycmathguy said:
The midpoint of a circle divides the diameter into two radii.
I asked about the mid-point of a diameter of a circle. How is the mid-point related to the center of the circle?
 
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  • #12
nycmathguy said:
The midpoint of a circle divides the diameter into two radii.
Quite futzin' around just DRAW it and get on with solving the problem. I don't understand why you have such an issue with follow through. You are going to keep having trouble with these problems if you don't get past that issue.
 
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  • #13
phinds said:
Quite futzin' around just DRAW it and get on with solving the problem. I don't understand why you have such an issue with follow through. You are going to keep having trouble with these problems if you don't get past that issue.
Follow through in what way? I'm stuck here.
 
  • #14
SammyS said:
I asked about the mid-point of a diameter of a circle. How is the mid-point related to the center of the circle?

Midpoint = [(11 + 3)/2, (-5 +15)/2]

Midpoint = (14/2, 10/2)

Midpoint = (7, 5)

This point = (h, k) = (7, 5).

So, h = 7 and k = 5.

I now plug into (x - h)^2 + (y - k)^2 = r^2.

Wait, I need the radius.

To find the radius, I must use the distance formula points.

So far, so good, yes?
 
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  • #15
phinds said:
Quite futzin' around just DRAW it and get on with solving the problem. I don't understand why you have such an issue with follow through. You are going to keep having trouble with these problems if you don't get past that issue.

I found h to 7 and k to be 5.

My friend completed the problem.

Here is her work:

distance between center (7,5) and point (3, 15) is the length of radius:

r=sqrt((7-3)^2+(5-15)^2)
r=sqrt(4^2+(-10)^2)
r=sqrt(16+100)
r=sqrt(116)

then r^2=116
(x - 7)^2 + (y - 5)^2 = 116=> your equation
 
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  • #16
nycmathguy said:
Follow through in what way? I'm stuck here.
Do you not understand how to put points on a graph?
 
  • #17
phinds said:
Do you not understand how to put points on a graph?
Forget it. Problem has been solved.
 
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