Finding an equation for a circle given two points

No idea how to do this, I used to have these conic shapes committed to memory but I forget them already.

I am supposed to find an equation for the circle that has center $(-1, 4)$ and passes through $(3, -2)$.

I tried graphing it, but it didn't help since the point was not in a straight line with the center and I have no idea what the diameter is.

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You aren't being given "two points" of the circle (that would be insufficient). You are being given the center and a point on the circle. Very different things. – Arturo Magidin Jan 1 '12 at 21:16

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I notice that you say "the point was not in a straight line with the center". But, given any two points, there is some straight line that joins them. (The straight line may happen to not be parallel to either of the coordinate axes.) – idmercer Jan 1 '12 at 23:40

6 Answers

Recall the:

Distance Formula:

The distance between the points $(x_1,y_1)$ and $(x_2,y_2)$ is $$D=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$

Example:

The radius of your circle is the distance between the points $(-1,4)$ and $(3,-2)$. Using the Distance Formula: $$D= \sqrt{ \bigl(3-(-1)\bigr)^2 + (-2-4)^2 } = \sqrt{ 4^2 + (-6)^2 } = \sqrt{ 16+36 } = \sqrt{ 52 } .$$

What is the equation of the circle?

It is important to realize the "equation of the circle" is: a point $(x,y)$ is on the circle if and only if the coordinates of the point $x$ and $y$ satisfy the equation. So, how to get the equation? What is the relationship between the $x$ and $y$ coordinates of a point on the circle?

Well, let $(x,y)$ be a point on the circle. The big idea is:

$$\text{The distance from the point }(x,y)\text{ to the center }(-1,4)\text{ is }\sqrt{52}.$$

So, using the distance formula (with $(x_2,y_2)=(x,y)$ and $(x_1,y_1)=(-1,4)$) , it follows that $$\sqrt{52}=\sqrt{\bigl(x-(-1)\bigr)^2 +(y-4)^2}.$$

Or $$52=(x+1)^2 +(y-4)^2.$$

The shortcut would be to just use the following formula (But it's important to realize why you'd use it and where it comes from):

Equation of a Circle

The equation of the circle with center located at $(a,b)$ and with radius $r$ is $$r^2=(x-a)^2 +(y-b)^2$$ Note that the radius squared is on the left-hand side of the equation.

The following may help:

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Am I supposed to have the distance formula memorized? – Jordan Jan 1 '12 at 21:32

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@Jordan Yes. (Not really, if all you want to do is find the equation of the circle. You can use the formula.) But it's really just the Pythagorean Theorem (draw a right triangle with the two points forming its hypotenuse). – David Mitra Jan 1 '12 at 21:37
I don't understand how this is the pythagorean theorem. I can work it on on a graph but I don't see how the equation fits that at all or why the answer isn't the square root of 52, not just 52. – Jordan Jan 1 '12 at 21:41
@Jordan I meant the distance formula is an application of the Pythagorean Theorem. Let $c$ be the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$. Call the distance between the $x$-coordinates $a$ and the distance between the $y$ coordinates $b$. Then $c^2=a^2+b^2$. And $a=|x_2-x_1|$, $b=|y_2-y_1|$. – David Mitra Jan 1 '12 at 21:44
@Jordan The radius of your circle is $\sqrt{52}$. The equation of the circle is $52=(x+1)^2+(y-4)^2$. – David Mitra Jan 1 '12 at 21:47
show 3 more comments

The equation of a circle with center $(a,b)$ and radius $r$ is $$(x-a)^2 + (y-b)^2 = r^2.$$

You know the center. You also know one point on the circle. The radius is the distance from the center of the circle to any one point on the circle.

So find the distance between $(-1,4)$ and $(3,-2)$ to get the radius. Then use the radius and the center to get the equation.

The diameter is, of course, twice the radius, but is irrelevant here.

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I don't know how to find the difference between two points on a graph. – Jordan Jan 1 '12 at 21:29
@Jordan: The distance between two points (which in this case, are not both "on a graph"), not the difference. I suspect you do know it, you just don't remember it or haven't made the connections. In any case, Wikipedia has the formula – Arturo Magidin Jan 1 '12 at 22:58

In general a circle with center $(h,k)$ and radius $r$ can be expressed as:

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

In the case of our questions, you have $(h,k)=(-1,4)$ in the above equation. You can then substitute $(-3,2)$ for $(x,y)$ to find $r$.

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Well you should know (or look up in your textbook) that a circle is described by the equation $(x - x_0)^2 + (y - y_0)^2 = r^2$ where $(x_0, y_0)$ is the center and $r$ is the radius. So you see your circle should be $(x +1)^2 + (y - 4)^2 = r^2$. The only thing missing is $r^2$. But the problem tells you (3, -2) is a point on the circle. So plugging this into your equation, you will find your answer.

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The equation of a circle with center $(a,b)$ and radius $r$ is $$(x-a)^2 + (y-b)^2 = r^2.$$

Because the center is $(a,b)=(-1,4)$ so $a=-1,b=4$. You also know one point on the circle. The radius is the distance from the center of the circle to any one point on the circle. The distance between $(-1,4)$ and $(3,-2)$ is $r=\sqrt52$ or $r^2=52$. Then use the radius and the center to get the equation.

$$(x+1)^2 + (y-4)^2 = 52$$

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Perhaps if you understand what a circle is, it may help. A circle is really just all points on the plane equidistant from the center.

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