Simplify this expression for an ellipse

[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!

 

[bigfooted]

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

 

[needingtoknow]

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

 

[qspeechc]

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?

 

[needingtoknow]

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!

 

[qspeechc]

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.

 

[Ray Vickson]

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 $4 f_1^2 f_2^2 = R^2$.

 

[needingtoknow]

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)

 

[qspeechc]

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

 

[needingtoknow]

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!