# Thread: Finding b-value of ellipse equation for given eccentricity

1. ## Finding b-value of ellipse equation for given eccentricity

Hi, this is my first time posting. I need help with this problem:

Suppose the equation for an ellipse is given by

$\frac{(x-3)^2}{4^2}+\frac{(y+2)^2}{b^2}=1$

Find $b$ so that the eccentricity is 0.75

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I know that eccentricity is defined as $\frac{c}{a}$, making $c=3$ and $a=4$. However, if I plug these values into the Pythagorean theorem, I get $b^2=-7$ (which is obviously incorrect, right? We're dealing with real numbers).

How do I solve this problem? Thanks in advance.

2. You ran into trouble when you went from $\frac{c}{a}=0.75$ to $c=3$ and $a=4$. You don't know the actual values of c and a, just the ratio (you could have chosen c=6 and a=8 instead, and $\frac{c}{a}$ would still be 0.75.

Try substituting $c=\sqrt{a^2-b^2}$. Then you can go two ways (and get two solutions). Either substitute $b=4$ or $a=4$. If you think about it, there should be two solutions, a "tall, skinny" ellipse and a "short, fat" ellipse.

So the first one will be $\sqrt{a^2-4^2}=0.75a$, and the second will be $\sqrt{4^2-b^2}=0.75*4$.

I think you can find the solutions from there. If you have any further questions, please post again in this thread.

- Hollywood

3. I'm not sure what you mean when you say substitute $c=\sqrt{a^2-b^2}$. Should I be substituting $a=4$ into the equation to get $c^2=4^2-b^2$? What do I do from there? I missed this day of class and am trying to figure out how to do this but my textbook gives no examples.

4. Originally Posted by bumpjump
Hi, this is my first time posting. I need help with this problem:

Suppose the equation for an ellipse is given by

$\frac{(x-3)^2}{4^2}+\frac{(y+2)^2}{b^2}=1$

Find $b$ so that the eccentricity is 0.75

-------------------------------------------

I know that eccentricity is defined as $\frac{c}{a}$, making $c=3$ and $a=4$. However, if I plug these values into the Pythagorean theorem, I get $b^2=-7$ (which is obviously incorrect, right? We're dealing with real numbers).

How do I solve this problem? Thanks in advance.
I will point you to here: The Ellipse ........................................

They have an example almost identical to your question!

5. Wow... I was wayyyy overthinking this problem. Thanks to both of you guys.