Expressing Geometric Properties with Equations - Common Core: High School - Geometry

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Question

What is the equation of the circle shown below?

Answer

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

If we look at the picture,

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

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

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Question

What is the equation of the circle shown below?

Answer

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

If we look at the picture,

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

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

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Question

What is the equation of the circle shown below?

Answer

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

If we look at the picture,

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

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

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Question

What is the equation of the circle shown below?

Answer

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

If we look at the picture,

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

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

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Question

What is the equation of the circle shown below?

Answer

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

If we look at the picture, we can see that the center is at $\left ( -4, -2\right )$

The next step it to find the radius. From looking at the picture,

we can see that the radius is 9.

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

$\left(x + 4\right)^{2} + \left(y + 2\right)^{2} = 81$

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Question

What is the equation of the circle shown below?

Answer

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

If we look at the picture,

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

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

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Question

What is the equation of the circle shown below?

Answer

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

If we look at the picture,

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

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

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Question

What is the equation of the circle shown below?

Answer

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

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

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

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Question

What is the equation of the circle shown below?

Answer

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

If we look at the picture,

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

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

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

We plug in the values, we get

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

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Question

What is the equation of the circle shown below?

Answer

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

If we look at the picture,

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

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

Compare your answer with the correct one above

Question

What is the equation of the circle shown below?

Answer

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

If we look at the picture,

we can see that the center is at $\left ( -2, -5\right )$

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

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

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Question

What is the equation of the circle shown below?

Answer

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

If we look at the picture,

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

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

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Question

Find the equation of an ellipse that has foci at and with a major axis distance of .

Answer

The general equation is

$\frac{1}{b^{2}} \left(- h + y\right)^{2} + \frac{1}{a^{2}} \left(x - 9\right)^{2} = 1$

$\left ( k, h\right )$ is the coordinates of the center of the foci.

is the distance to the foci on the x-axis from the center, and is the distance to the foci from the y-axis.

The first step is to figure out what the focal radii is .

Since we know the distance of the major axis, all we need to do is set up a simple equation.

Now solve for

The next step is to find the center between the foci.

To find the center, we simply find the average between the coordinates.

$\x = 2 \y = 3.0$

So the center of the foci is at $\left ( 2, 3\right )$ .

Now we need to figure out what the distance from the center to the foci is .

We do this by taking the x-coordinate from one of the foci and subtracting it from the center x-coordinate

Now the last part is to find .

We can find it by using the following equation.

$b^{2} = a^{2} - c^{2}$

We simply substitute with 5.0 and with -7

$b^{2} = -24.0$

Now we can substitute these values into the general equation to get.

$\frac{1}{25} \left(x - 2\right)^{2} - \frac{1}{24} \left(y - 3\right)^{2} = 1$

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Question

Find the equation of an ellipse that has foci at and with a major axis distance of .

Answer

The general equation is

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

$\left ( k, \quad h\right )$ is the coordinates of the center of the foci.

is the distance to the foci on the x-axis from the center, and is the distance to the foci from the y-axis.

The first step is to figure out what the focal radii is .

Since we know the distance of the major axis, all we need to do is set up a simple equation.

Now solve for .

The next step is to find the center between the foci.

To find the center, we simply find the average between the coordinates.

$\x = 0 \y = 1.0$

So the center of the foci is at $\left ( 0, \quad 1\right )$

Now we need to figure out what the distance from the center to the foci is .

We do this by taking the x-coordinate from one of the foci and subtracting it from the center x-coordinate

Now the last part is to find .

We can find it by using the following equation.

$b^{2} = a^{2} - c^{2}$

We simply substitute with 9.0 and with -7

$b^{2} = 32.0$

Now we can substitute these values into the general equation to get.

$\frac{x^{2}}{81} + \frac{1}{32} \left(y - 1\right)^{2} = 1$

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Question

Find the equation of an ellipse that has foci at and with a major axis distance of .

Answer

The general equation is

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

$\left ( k, \quad h\right )$ is the coordinates of the center of the foci.

is the distance to the foci on the x-axis from the center, and is the distance to the foci from the y-axis.

The first step is to figure out what the focal radii is.

Since we know the distance of the major axis, all we need to do is set up a simple equation.

Now solve for

The next step is to find the center between the foci.

To find the center, we simply find the average between the coordinates.

$\x = -1 \y = 4.0$

So the center of the foci is at $\left ( -1, \quad 4\right )$

Now we need to figure out what the distance from the center to the foci is.

We do this by taking the x-coordinate from one of the foci and subtracting it from the center x-coordinate

Now the last part is to find .

We can find it by using the following equation.

$b^{2} = a^{2} - c^{2}$

We simply substitute with 9.0 and with -4

$b^{2} = 65.0$

Now we can substitute these values into the general equation to get.

$\frac{1}{81} \left(x + 1\right)^{2} + \frac{1}{65} \left(y - 4\right)^{2} = 1$

Compare your answer with the correct one above

Question

Find the equation of an ellipse that has foci at and with a major axis distance of .

Answer

The general equation is

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

$\left ( k, \quad h\right )$ is the coordinates of the center of the foci.

is the distance to the foci on the x-axis from the center, and is the distance to the foci from the y-axis.

The first step is to figure out what the focal radii is.

Since we know the distance of the major axis, all we need to do is set up a simple equation.

Now solve for

The next step is to find the center between the foci.

To find the center, we simply find the average between the coordinates.

$\x = -2 \ y = 5.0$

So the center of the foci is at $\left ( -2, \quad 5\right )$

Now we need to figure out what the distance from the center to the foci is .

We do this by taking the x-coordinate from one of the foci and subtracting it from the center x-coordinate.

Now the last part is to find .

We can find it by using the following equation.

$b^{2} = a^{2} - c^{2}$

We simply substitute with 8.0 and with 1

$b^{2} = 63.0$

Now we can substitute these values into the general equation to get.

$\frac{1}{64} \left(x + 2\right)^{2} + \frac{1}{63} \left(y - 5\right)^{2} = 1$

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Question

Find the equation of an ellipse that has foci at and with a major axis distance of .

Answer

The general equation is

$\frac{1}{b^{2}} \left(- h + y\right)^{2} + \frac{1}{a^{2}} \left(x - 4\right)^{2} = 1$

$\left ( k, \quad h\right )$ is the coordinates of the center of the foci.

is the distance to the foci on the x-axis from the center, and is the distance to the foci from the y-axis.

The first step is to figure out what the focal radii is.

Since we know the distance of the major axis, all we need to do is set up a simple equation.

Now solve for

The next step is to find the center between the foci.

To find the center, we simply find the average between the coordinates.

$\x = 1 \y = -10.0$

So the center of the foci is at $\left ( 1, \quad -10\right )$

Now we need to figure out the distance from the center to the foci .

We do this by taking the x-coordinate from one of the foci and subtracting it from the center x-coordinate

Now the last part is to find .

We can find it by using the following equation.

$b^{2} = a^{2} - c^{2}$

We simply substitute with 1.0 and with -3.

$b^{2} = -8.0$

Now we can substitute these values into the general equation to get.

$\left(x - 1\right)^{2} - \frac{1}{8} \left(y + 10\right)^{2} = 1$

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Question

Find the equation of an ellipse that has foci at and with a major axis distance of .

Answer

The general equation is

$\frac{1}{b^{2}} \left(- h + y\right)^{2} + \frac{1}{a^{2}} \left(x - 5\right)^{2} = 1$

$\left ( k, \quad h\right )$ is the coordinates of the center of the foci.

is the distance to the foci on the x-axis from the center, and is the distance to the foci from the y-axis.

The first step is to figure out what the focal radii is.

Since we know the distance of the major axis, all we need to do is set up a simple equation.

Now solve for

The next step is to find the center between the foci.

To find the center, we simply find the average between the coordinates.

$\x = \frac{15}{2} \\ y = 6.0$

So the center of the foci is at $\left ( \frac{15}{2}, \quad 6\right )$

Now we need to figure out what the distance from the center to where the foci is.

We do this by taking the x-coordinate from one of the foci and subtracting it from the center x-coordinate

$c = \frac{5}{2}$

Now the last part is to find .

We can find it by using the following equation.

$b^{2} = a^{2} - c^{2}$

We simply substitute with 9.0 and with $\frac{5}{2}$

$b^{2} = 74.75$

Now we can substitute these values into the general equation to get.

$\frac{1}{81} \left(x - \frac{15}{2}\right)^{2} + \frac{4}{299} \left(y - 6\right)^{2} = 1$

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Question

Find the equation of an ellipse that has foci at and with a major axis distance of .

Answer

The general equation is

$\frac{1}{b^{2}} \left(- h + y\right)^{2} + \frac{1}{a^{2}} \left(x + 10\right)^{2} = 1$

$\left ( k, \quad h\right )$ is the coordinates of the center of the foci.

is the distance to the foci on the x-axis from the center, and is the distance to the foci from the y-axis.

The first step is to figure out what the focal radii is.

Since we know the distance of the major axis, all we need to do is set up a simple equation.

Now solve for

The next step is to find the center between the foci.

To find the center, we simply find the average between the coordinates.

$\x = -3 \y = -10.0$

So the center of the foci is at $\left ( -3, \quad -10\right )$

Now we need to figure out what the distance from the center to the foci is.

We do this by taking the x-coordinate from one of the foci and subtracting it from the center x-coordinate

Now the last part is to find .

We can find it by using the following equation.

$b^{2} = a^{2} - c^{2}$

We simply substitute with 6.0 and with 7

$b^{2} = -13.0$

Now we can substitute these values into the general equation to get.

$\frac{1}{36} \left(x + 3\right)^{2} - \frac{1}{13} \left(y + 10\right)^{2} = 1$

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Question

Find the equation of an ellipse that has foci at and with a major axis distance of .

Answer

The general equation is

$\frac{1}{b^{2}} \left(- h + y\right)^{2} + \frac{1}{a^{2}} \left(x - 5\right)^{2} = 1$

$\left ( k, \quad h\right )$ is the coordinates of the center of the foci.

is the distance to the foci on the x-axis from the center, and is the distance to the foci from the y-axis.

The first step is to figure out what the focal radii is ().

Since we know the distance of the major axis, all we need to do is set up a simple equation.

Now solve for

The next step is to find the center between the foci.

To find the center, we simply find the average between the coordinates.

$\x = - \frac{1}{2} \\y = -1.0$

So the center of the foci is at $\left ( - \frac{1}{2}, \quad -1\right )$

Now we need to figure out what the distance from the center to the foci .

We do this by taking the x-coordinate from one of the foci and subtracting it from the center x-coordinate

$c = - \frac{11}{2}$

Now the last part is to find .

We can find it by using the following equation.

$b^{2} = a^{2} - c^{2}$

We simply substitute with 7.0 and with $- \frac{11}{2}$

$b^{2} = 18.75$

Now we can substitute these values into the general equation to get.

$\frac{1}{49} \left(x + \frac{1}{2}\right)^{2} + \frac{4}{75} \left(y + 1\right)^{2} = 1$

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