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# Circle equations — Harder example

​Watch Sal work through a harder Circle equations problem.

## Want to join the conversation?

• Why h and k? Standard way of doing it?
• I imagine that it's arbitrary. Usually, you use x and y, but since they are variables and the numbers are given, they are using h and k as constants.
• How do u find the radius if it not simple like this problem?
• You can always calculate the distance between the center and the point lying on the circumference (ie. the radius) by using this formula:
[(x-h)^2 + (y-k)^2]^1/2
For more info you can check out this website. It's very useful:
http://www.mathwarehouse.com/algebra/distance_formula/index.php
Hope this helps! :-)
• what is the circumference of a circle whose area is 100 pie
• 20 pie. The equation for the area of the circle is A= (pie)(radius^2). Using that, you can calculate that the radius of the circle is 10. The equation for the circumference of a circle is C=2(pie)(radius) or C=(pie)(diameter). Plug the radius of 10 into an equation and you should find the circumference of the circle to be 20 pie.
• HERE IS THE PROVIDED QUESTION:
"A circle in the xy-plane contains the points (−1,1), (1,1), and (−1,−1). Which of the following is an equation of the circle?"

The formula for a circle with a center (h,k) is:
(x−h)^2+(y−k)^2=r^2

We could check which formula is satisfied by the three given points.
Alternatively, let's substitute the three points for (x,y)

First, substituting (1,1), we have:
(1−h)^2+(1−k)^2=r^2

Substituting (−1,1), we have:
(−1−h)^2+(1−k)^2=r^2

Finally, substituting (−1,−1), we have:
(−1−h)^2+(−1−k)^2=r^2

Next, let's subtract the second equation from the first. We get:
(1−h)^2−(−1−h)^2=0
OR
h^2−2h+1−(h^2+2h+1)=0
−4h=0
h=0

Let's subtract the third equation from the second. We get:
(1−k)^2−(−1−k)^2=0

This is the same equation as that above with
k instead of h. So k=0, and the center of the circle is (0,0). Now, let's find r.

Since (1,1) is a point on the circle, and the center is (0,0), the radius is squarerootof2.

The equation of the circle is
x^2+y^2=2x OR x^2+y^2−2=0

HERE ARE MY QUESTIONS:

I don't understand how the answer was found, and I also have a few general questions:
1. Is it assumed that none three given points are the center of the circle?
Because I do not know how to write a circle equation without knowing the center. If the center is known to not be one of the three listed points, then it is obviously (0,0), but if the center IS one of the listed points, then it is (-1,1). However, I do not know if the center is included in the listed points or not, so I couldn't figure much else out without that information....

2. The provided answer talks about subtracting equations from each other like it's a normal thing. I have never encountered a situation where an equation is subtracted form another when dealing with circles... elaborate?
• Yes, if a point lies ON the circle, then it cannot be its center. That point must be on the circle's edge. I would suggest imagining the coordinates in your head, since the provided points on the circle were very simple. It wouldn't be too difficult to visualize that the center must be (0,0) from the given points.
• So...based on this video, can it be assumed that when the SAT says "a point on a circle," it is the radius? Like in this case it was 20?
• A point on the circle is literally any point lying on the circumference. To get the radius, you would have to calculate the distance between that point and the circle's center. A point on the circle does NOT mean radius.
Hope this helps.
• Best of luck for today everyone ! Do your best !!
• What if the x co-ordinates are different ?