# Analytic geometry question

Find the vertices, foci, and eccentricity of the ellipse.

1/2x^2 + 1/8y^2 = 1/4

i'm a little confused on how to clear the fraction and set the equation up to make it equal 1.
I tried to divide everything by 4 to clear it but 8 cant divide into 4 and if I use a higher number than 4 the equation wont equal 1. when the fractions are cleared the equation should be in the form of x^2/a^2 + y^2/b^2 = 1 if the equation is horizontal or x^2/b^2 + y^2/a^2 = 1 if it is vertical.
Can someone help me clear the fractions?

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cepheid
Staff Emeritus
Gold Member
The right hand side is equal to 1/4. So, in order to make it equal 1, you would have to multiply it by 4, not divide. Of course, anything you do to one side of the equation, you must do to the other side. Therefore, you end up with

4(1/2)x2 + 4(1/8)y2 = 4/4

Can you take it from here?

SteamKing
Staff Emeritus
Homework Helper
If you are trying to clear fractions, division is not the recommended approach.

I have x^2/2 + y^2/1/2 = 1. I get how you told me to this but im still a little confused on why you multiplied the left side by 4 and then divided the right side to make it equal 1. why wouldnt you multiply the right side like the left and get 16? (I know your way is right im just trying to fully get it)

cepheid
Staff Emeritus
Gold Member
I have x^2/2 + y^2/1/2 = 1. I get how you told me to this but im still a little confused on why you multiplied the left side by 4 and then divided the right side to make it equal 1. why wouldnt you multiply the right side like the left and get 16? (I know your way is right im just trying to fully get it)
Both sides are being multiplied by 4. For the right-hand side:

(1/4) * 4 = 1

I don't know where you get 16 from.

EDIT: oh, and by the way, your left side is wrong as well. Look at the coefficient of the x2 term:

4*(1/2) = 4/2 = ?

As for the coefficient of the y2 term:

4*(1/8) = 4/8 = ?

You fill in the question marks.

Last edited:
doesnt 4/8 = .5 or 1/2?

eumyang
Homework Helper
Find the vertices, foci, and eccentricity of the ellipse.

1/2x^2 + 1/8y^2 = 1/4
Can you please use parentheses? One could interpret the above as
$$\frac{1}{2x^2} + \frac{1}{8y^2} = \frac{1}{4}$$
... which would be wrong.

doesnt 4/8 = .5 or 1/2?
So now you have
$$\frac{x^2}{2} + \frac{y^2}{8} = \frac{1}{4}$$
$$\rightarrow 2x^2 + \frac{y^2}{2} = 1$$

Now, rewrite the first term as a fraction.
$$2x^2 = \frac{x^2}{\text{?}}$$

cepheid
Staff Emeritus
Gold Member
doesnt 4/8 = .5 or 1/2?
Yes, so the coefficient multiplying the y2 term is 1/2 i.e.:

(1/2)y2

That's not what you wrote though. You had y2 divided by 1/2 (at least, I think so, although there is some ambiguity in what you wrote due to the lack of parentheses).

Anyways, so as eumyang said, you now have:

2x2 + (1/2)y2 = 1

and you need to write these coefficients in the form (1/a2) and (1/b2). I.e.

2 = (1/a2)

a2 = ?

and,

1/2 = (1/b2)

b2 = ?