Geometry - Derive Circle Equation: CCSS.Math.Content.HSG-GPE.A.1

What is the equation of the circle shown below?
Plot2

$\left(x + 9\right)^{2} + \left(y + 5\right)^{2} = 36$

$\left(x + 9\right)^{2} + \left(y + 5\right)^{2} = 6$

$\left(x - 9\right)^{2} + \left(y + 5\right)^{2} = 36$

$\left(x + 9\right)^{2} + \left(y - 5\right)^{2} = 6$

$\left(x - 9\right)^{2} + \left(y - 5\right)^{2} = 36$

Explanation

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,

Plot2

we can see that the center is at $\left ( -9, -5\right )$ .

The next step it to find the radius. Recall the radius is the distance from the center of the circle to any point of the circle's edge.

From looking at the picture, we can see that the radius is 6.

With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ .

When we plug in the values, we get

$\left(x + 9\right)^{2} + \left(y + 5\right)^{2} = 36$

What is the equation of the circle shown below?

Plot3

$\left(x + 9\right)^{2} + \left(y + 2\right)^{2} = 36$

$\left(x + 9\right)^{2} + \left(y + 2\right)^{2} = 6$

$\left(x - 9\right)^{2} + \left(y + 2\right)^{2} = 36$

$\left(x + 9\right)^{2} + \left(y - 2\right)^{2} = 6$

$\left(x - 9\right)^{2} + \left(y - 2\right)^{2} = 36$

Explanation

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,

Plot3

we can see that the center is at $\left ( -9, -2\right )$ .

The next step it to find the radius From looking at the picture, we can see that the radius is 6.

With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ .

We plug in the values, we get

$\left(x + 9\right)^{2} + \left(y + 2\right)^{2} = 36$

What is the equation of the circle shown below?
Plot6

$\left(x + 3\right)^{2} + \left(y - 3\right)^{2} = 1$

$\left(x - 3\right)^{2} + \left(y - 3\right)^{2} = 1$

$\left(x + 3\right)^{2} + \left(y + 3\right)^{2} = 1$

$\left(x - 3\right)^{2} + \left(y + 3\right)^{2} = 1$

Explanation

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,

Plot6

we can see that the center is at $\left ( -3, 3\right )$

The next step it to find the radius From looking at the picture, we can see that the radius is 1.

With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$

We plug in the values, we get

$\left(x + 3\right)^{2} + \left(y - 3\right)^{2} = 1$

What is the equation of the circle shown below?
Plot7

$\left(x - 2\right)^{2} + \left(y - 2\right)^{2} = 16$

$\left(x - 2\right)^{2} + \left(y - 2\right)^{2} = 4$

$\left(x + 2\right)^{2} + \left(y - 2\right)^{2} = 16$

$\left(x - 2\right)^{2} + \left(y + 2\right)^{2} = 4$

$\left(x + 2\right)^{2} + \left(y + 2\right)^{2} = 16$

Explanation

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,

Plot7

we can see that the center is at $\left ( 2, 2\right )$

The next step it to find the radius From looking at the picture, we can see that the radius is 4.

With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$

We plug in the values, we get

$\left(x - 2\right)^{2} + \left(y - 2\right)^{2} = 16$

What is the equation of the circle shown below?

Plot4

$\left(x + 10\right)^{2} + \left(y - 4\right)^{2} = 16$

$\left(x + 10\right)^{2} + \left(y - 4\right)^{2} = 4$

$\left(x - 10\right)^{2} + \left(y - 4\right)^{2} = 16$

$\left(x + 10\right)^{2} + \left(y + 4\right)^{2} = 4$

$\left(x - 10\right)^{2} + \left(y + 4\right)^{2} = 16$

Explanation

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,

Plot4

we can see that the center is at $\left ( -10, 4\right )$

The next step it to find the radius From looking at the picture, we can see that the radius is 4.

With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$

We plug in the values, we get

$\left(x + 10\right)^{2} + \left(y - 4\right)^{2} = 16$

What is the equation of the circle shown below?
Plot1

$\left(x + 7\right)^{2} + \left(y + 2\right)^{2} = 16$

$\left(x + 7\right)^{2} + \left(y + 2\right)^{2} = 4$

$\left(x - 7\right)^{2} + \left(y + 2\right)^{2} = 16$

$\left(x + 7\right)^{2} + \left(y - 2\right)^{2} = 4$

$\left(x - 7\right)^{2} + \left(y - 2\right)^{2} = 16$

Explanation

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,

Plot1

we can see that the center is at

$\left ( -7, -2\right )$

The next step it to find the radius. Recall the radius is the distance from the center of the circle to any point on the edge of the circle.

From looking at the picture, we can see that the radius is 4. With this information, we can plug it into the general circle equation

The general circle equation is,

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ . When we plug in the values, we get

$\left(x + 7\right)^{2} + \left(y + 2\right)^{2} = 16$

What is the equation of the circle shown below?

Plot12

$\left(x + 2\right)^{2} + \left(y - 2\right)^{2} = 4$

$\left(x + 2\right)^{2} + \left(y - 2\right)^{2} = 2$

$\left(x - 2\right)^{2} + \left(y - 2\right)^{2} = 4$

$\left(x + 2\right)^{2} + \left(y + 2\right)^{2} = 2$

$\left(x - 2\right)^{2} + \left(y + 2\right)^{2} = 4$

Explanation

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,
Plot12
we can see that the center is at $\left ( -2, 2\right )$ .

The next step it to find the radius. From looking at the picture, we can see that the radius is 2.

With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$
We plug in the values, we get

$\left(x + 2\right)^{2} + \left(y - 2\right)^{2} = 4$

What is the equation of the circle shown below?

Plot10

$\left(x + 5\right)^{2} + \left(y - 6\right)^{2} = 4$

$\left(x + 5\right)^{2} + \left(y - 6\right)^{2} = 2$

$\left(x - 5\right)^{2} + \left(y - 6\right)^{2} = 4$

$\left(x + 5\right)^{2} + \left(y + 6\right)^{2} = 2$

$\left(x - 5\right)^{2} + \left(y + 6\right)^{2} = 4$

Explanation

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,
Plot10
we can see that the center is at $\left ( -5, 6\right )$

The next step it to find the radius. From looking at the picture, we can see that the radius is 2. With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$

We plug in the values, we get

$\left(x + 5\right)^{2} + \left(y - 6\right)^{2} = 4$

What is the equation of the circle shown below?

Plot8

$\left(x + 10\right)^{2} + \left(y + 3\right)^{2} = 36$

$\left(x + 10\right)^{2} + \left(y + 3\right)^{2} = 6$

$\left(x - 10\right)^{2} + \left(y + 3\right)^{2} = 36$

$\left(x + 10\right)^{2} + \left(y - 3\right)^{2} = 6$

$\left(x - 10\right)^{2} + \left(y - 3\right)^{2} = 36$

Explanation

In order to find the equation, we must find the coordinates of the center of the circle If we look at the picture,

Plot8

we can see that the center is at $\left ( -10, -3\right )$

The next step it to find the radius. From looking at the picture, we can see that the radius is 6. With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ .

We plug in the values, we get

$\left(x + 10\right)^{2} + \left(y + 3\right)^{2} = 36$

What is the equation of the circle shown below?

Plot9

$\left(x - 1\right)^{2} + \left(y - 7\right)^{2} = 4$

$\left(x - 1\right)^{2} + \left(y - 7\right)^{2} = 2$

$\left(x + 1\right)^{2} + \left(y - 7\right)^{2} = 4$

$\left(x - 1\right)^{2} + \left(y + 7\right)^{2} = 2$

$\left(x + 1\right)^{2} + \left(y + 7\right)^{2} = 4$

Explanation

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,

Plot9

we can see that the center is at $\left ( 1, 7\right )$

The next step it to find the radius. From looking at the picture, we can see that the radius is 2. With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$