求圆的圆心和半径

(Video) General Form

by mathman1024

Here's a video by mathman1024 showing you how to turn an equation in general form into an equation in center-radius form so you can find the center and radius of a circle. The general form of the circle will be: $$x^2+y^2+cx+dy+e=0$$ To turn this into something we can understand we have to complete the square twice: once for the $x$ and once for the $y$.

The first problem he looks at is: $x^2+y^2+10x-14y+70=0$. The first step is to put the constant on the other side and rearrange the variable terms. $$\begin{align} &x^2+y^2+10x-14y+70=0\\ &x^2+y^2+10x-14y+70\color{#8925AE}{-70}=0\color{#8925AE}{-70}\\ &x^2+10x+y^2-14y=-70 \end{align}$$ Now we complete the square for both $x$ and $y$. Remember to do this we take the $x$ or $y$ term, half it, square it, and add the squared terms to both sides. $$\begin{align} &x^2+10x+y^2-14y=-70\\ &x^2+10x\color{#8925AE}{+\bigg(\frac{10}{2}\bigg)^2}+y^2-14y\color{#8925AE}{+\bigg(\frac{-14}{2}\bigg)^2}\\ &=-70\color{#8925AE}{+\bigg(\frac{10}{2}\bigg)^2+\bigg(\frac{-14}{2}\bigg)^2}\\ &x^2+10x+25+y^2-14y+49=4\\ &(x+5)^2+(y-7)^2=4 \end{align}$$ Now that our equation is in center-radius form we can find the center and the radius. $$\text{Center: }(-5,7)\\ r=2$$

He does one more example. Try this one on your own then see if you get the same answer.

1. $x^2+y^2-6x-4y-12=0$

Finding the Center and Radius of a Circle

There are two ways you can find the center and radius of a circle: by graphing or reading an equation. We will start with reading an equation.

Reading an Equation

As we saw in the previous lesson, circles often follow a standard form. We also learned how to convert from different forms. This lesson will help you to find the equation of a circle using the radius and center. The following equation is the standard form of a circle:

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

You already have the tools to find the center and radius using the standard form! Radius, $r$, is the simplest. Just take the square root of the right-hand side ($r^2$) of the equation in standard form.

The center is a tad bit more complicated, but nothing you can't handle. To find the center of a circle using standard form, know that the circle's center is found at $(h,k)$. Beware of the signs!

Reading a Graph

To find the center and radius of a circle that is graphed requires a little deduction. Let's start with the radius. We know that the radius is one-half of the diameter of a circle. We also know that the diameter is the distance from one point of the circle to the absolute other side of the circle.

In the above graph, we can see two points on the circle labeled, one at $(-3,4)$ and one at $(7,4)$. Now, we need to calculate the distance between the two points in order to find the diameter. Since the $y$ values of both points are the same, we need only to consider the $x$ values. The distance between the two points is $10$.

Therefore, the radius of the circle is $5$.

Next, we will find the center of the circle. If we go back to the definition of a circle, we know that the center of a circle is a point equidistance from any point on the circle. Another way to think of the center of a circle is to look at the halfway point on a diameter of the circle. We can continue to consider the above graph.

The halfway point between both points can be found at $(2,4)$.

The Center and Radius of a Circle

We know that a circle is a shape made up of all points that are the same distance away from a central point. That central point is called the center of the circle. The distance from the center to any point on the circle is called the radius.

We can always use the center and radius to graph the circle.

We can write the equation of a circle using the standard form:

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

In this equation, the center is $(h,k)$, and the radius is $r$. Notice the values for the center are opposite to what the numbers are in the vertex.

The radius will always be a positive because it's essentially a distance - and distance can never be positive.

If you are given a circle's graph, you can find the center by looking at the middle of the circle. You can count how far it goes from the center to the edge to find the radius. If you are given an equation, just match it to the standard form and identify $h$, $k$, and $r$.

Knowing the center and radius helps you to graph the circle or write its equation. Circles are all about symmetry and distance - and once you know the center and radius, you know everything about the circle.