# Simplify this expression for an ellipse

• needingtoknow
In summary, the equation sqrt(x^2+(y-sqrt(5))^2) = 8 - sqrt(x^2 + (y+sqrt(5))^2) can be simplified by squaring both sides twice and rearranging the terms to get the equation 16x^2 + 11y^2 = 176.
needingtoknow

## Homework Statement

Simplify sqrt(x^2+(y-sqrt(5))^2) = 8 - sqrt(x^2 + (y+sqrt(5))^2)

## The Attempt at a Solution

I know squaring both sides, collecting like terms and simplifying gets the equation but in my solution manual they do it a different way that is a lot shorter and I need help understanding what they did:

sqrt(x^2+(y-sqrt(5))^2) = 8 - sqrt(x^2 + (y+sqrt(5))^2)
16 + y*sqrt(5) = 4*sqrt(x^2 +(y+sqrt(5))^2)
from this step ^^ to the following is where I am having trouble
16x^2 + 11y^2 = 176

Any help will be greatly appreciated. Thank you!

They square the left hand side and right hand side again.

But if you square the left and right side won't the trinomial on the right side expand to something absurbdly long?

On the right is four times the square root of something, so when you square it you get 16(x^2 +(y+sqrt(5))^2)

What's the problem?

Oh yes sorry I believe I pointed to the wrong line. I am having trouble going from this line:

sqrt(x^2+(y-sqrt(5))^2) = 8 - sqrt(x^2 + (y+sqrt(5))^2)

to this line:

16 + y*sqrt(5) = 4*sqrt(x^2 +(y+sqrt(5))^2)

Sorry for the mistake and thank you again!

After squaring both sides once you get
$$x^2+(y-\sqrt{5})^2=64-16\sqrt{x^2+(y+\sqrt{5})^2}+x^2+(y+\sqrt{5})^2$$

Then put the square root on one side on its own, and all the other terms on the other side, and square again.

needingtoknow said:

## Homework Statement

Simplify sqrt(x^2+(y-sqrt(5))^2) = 8 - sqrt(x^2 + (y+sqrt(5))^2)

## The Attempt at a Solution

I know squaring both sides, collecting like terms and simplifying gets the equation but in my solution manual they do it a different way that is a lot shorter and I need help understanding what they did:

sqrt(x^2+(y-sqrt(5))^2) = 8 - sqrt(x^2 + (y+sqrt(5))^2)
16 + y*sqrt(5) = 4*sqrt(x^2 +(y+sqrt(5))^2)
from this step ^^ to the following is where I am having trouble
16x^2 + 11y^2 = 176

Any help will be greatly appreciated. Thank you!

For ##f_1 = \sqrt{x^2+(y-\sqrt{5})^2}## and ##f_2 = \sqrt{x^2+(y+\sqrt{5})^2}## we can write the equation as ##f_1+f_2 = 8##, hence ##64 = (f_1 + f_2)^2 = f_1^2 + f_2^2 + 2 f_1 f_2##. Re-write this as ##2 f_1 f_2 = 64 - f_1^2 - f_2^2 \equiv R##, and note that when you expand out and simplify the right-hand-side R it does not contain any square roots at all. Square again to get ##4 f_1^2 f_2^2 = R^2##.

Yes, that is exactly what I did, but I wanted to know how to get this line that was given in the solution manual from the original expression:

16 + y*sqrt(5) = 4*sqrt(x^2 +(y+sqrt(5))^2)

Can you show us your next steps so we can see where you went wrong?

Actually I understand what you meant. I redid the problem square it twice like you said and got the desired equation. Thank you all for your help and sorry for the confusion!

## 1. What is an ellipse?

An ellipse is a closed curve that is formed by the intersection of a cone and a plane that does not pass through the apex of the cone. It is often described as a "stretched out" circle.

## 2. How do you simplify an expression for an ellipse?

To simplify an expression for an ellipse, you need to use the standard form of an ellipse equation, which is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes, respectively.

## 3. What is the difference between an ellipse and a circle?

A circle is a special case of an ellipse where both the major and minor axes are equal in length. This means that the standard form equation for a circle is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle and r is the radius. In other words, a circle is a perfectly symmetrical ellipse.

## 4. Can an ellipse have a negative value for the semi-major or semi-minor axes?

Yes, an ellipse can have negative values for the semi-major and semi-minor axes. This just means that the ellipse is rotated or flipped in relation to the x and y axes. The standard form equation can still be used, but the values for a and b will be negative.

## 5. How can simplifying an ellipse expression be useful?

Simplifying an ellipse expression can be useful in many ways, such as graphing the ellipse, finding important points like the foci and vertices, and solving problems involving ellipses in real life applications, such as orbits of planets or satellites.

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