Common Core: High School - Geometry : Derive Ellipse and Hyperbola Equations: CCSS.Math.Content.HSG-GPE.A.3

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Example Question #1 : Derive Ellipse And Hyperbola Equations: Ccss.Math.Content.Hsg Gpe.A.3

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

 

Possible Answers:

Correct answer:

Explanation:

The general equation is

 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.

So the center of the foci is at .

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.

We simply substitute  with 5.0 and  with -7

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

 

 

Example Question #2 : Derive Ellipse And Hyperbola Equations: Ccss.Math.Content.Hsg Gpe.A.3

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

 

Possible Answers:

Correct answer:

Explanation:

The general equation is

 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.

So the center of the foci is at 

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.

We simply substitute  with 9.0 and  with -7

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

 

 

Example Question #3 : Derive Ellipse And Hyperbola Equations: Ccss.Math.Content.Hsg Gpe.A.3

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

 

Possible Answers:

Correct answer:

Explanation:

The general equation is

 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.

So the center of the foci is at 

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.

We simply substitute  with 9.0 and  with -4

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

 

 

Example Question #4 : Derive Ellipse And Hyperbola Equations: Ccss.Math.Content.Hsg Gpe.A.3

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

 

Possible Answers:

Correct answer:

Explanation:

The general equation is

 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.

So the center of the foci is at 

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.

We simply substitute  with 8.0 and  with 1

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

 

 

Example Question #5 : Derive Ellipse And Hyperbola Equations: Ccss.Math.Content.Hsg Gpe.A.3

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

 

Possible Answers:

Correct answer:

Explanation:

The general equation is

 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.

So the center of the foci is at

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.

We simply substitute  with 1.0 and  with -3.

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

 

 

Example Question #6 : Derive Ellipse And Hyperbola Equations: Ccss.Math.Content.Hsg Gpe.A.3

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

 

Possible Answers:

Correct answer:

Explanation:

The general equation is

 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.

So the center of the foci is at 

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

Now the last part is to find .

We can find it by using the following equation.

We simply substitute  with 9.0 and  with 

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

 

 

Example Question #7 : Derive Ellipse And Hyperbola Equations: Ccss.Math.Content.Hsg Gpe.A.3

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

 

Possible Answers:

Correct answer:

Explanation:

The general equation is

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.

So the center of the foci is at 

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.

We simply substitute  with 6.0 and  with 7

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

 

 

Example Question #8 : Derive Ellipse And Hyperbola Equations: Ccss.Math.Content.Hsg Gpe.A.3

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

Possible Answers:

 

Correct answer:

Explanation:

The general equation is

 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.

So the center of the foci is at 

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

Now the last part is to find .

We can find it by using the following equation.

We simply substitute  with 7.0 and  with 

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

 

 

Example Question #9 : Derive Ellipse And Hyperbola Equations: Ccss.Math.Content.Hsg Gpe.A.3

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

 

 

Possible Answers:

Correct answer:

Explanation:

The general equation is

 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.

So the center of the foci is at 

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.

We simply substitute  with 9.0 and  with 

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

 

 

Example Question #10 : Derive Ellipse And Hyperbola Equations: Ccss.Math.Content.Hsg Gpe.A.3

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

Possible Answers:

Correct answer:

Explanation:

The general equation is

 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.

So the center of the foci is at

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.

We simply substitute  with 8.0 and  with 

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

 

 

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All Common Core: High School - Geometry Resources

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