1. ## Graphing Off-Center Conics

I'm a bit lost at the moment

I was unfortunately absent on the day of graphing off-center conics in our Algebra 2 class. I've got the notes, but the teacher refused to teach me the subject before or after class. I'm going in tomorrow before school to talk with her, but in the meantime, I could use some help with this.

For instance;

How do I write an equation and graph the ellipse with foci at (4,2) and vertices at (4,4) and (4,-8) ?

I understand standard form, but I don't understand how to add the center points.

2. Hello ScottStedman

Welcome to Math Help Forum!
Originally Posted by ScottStedman
I'm a bit lost at the moment

I was unfortunately absent on the day of graphing off-center conics in our Algebra 2 class. I've got the notes, but the teacher refused to teach me the subject before or after class. I'm going in tomorrow before school to talk with her, but in the meantime, I could use some help with this.

For instance;

How do I write an equation and graph the ellipse with foci at (4,2) and vertices at (4,4) and (4,-8) ?

I understand standard form, but I don't understand how to add the center points.

If the centre of the ellipse is at $(h, k)$ then the equation is:

$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$

where $2a$ and $2b$ are, as usual, the lengths of the axes. This is simply because $(x-h) = 0$ when $x =h$ and $(y-k) = 0$ when $y = k$, so the centre has shifted from $(0,0)$ to $(h,k)$.

In the example you've given, the centre is at the mid-point of the line joining $(4,4)$ to $(4,-8)$; i.e. at $(4,-2)$. So $h = 4, k =-2$.

The length of the vertical axis is $12$ (the distance between these vertices), so $b=6$.

The foci always lie on the major axis, so since one focus is at $(4,2)$, the major axis is vertical, and the foci are at a distance $be$ from the centre. So $be = 4$. And, when the major axis is vertical, $a^2 = b^2-b^2e^2 = 36 - 16=20$.

Putting all this together, I reckon the equation is:

$\frac{(x-4)^2}{20} + \frac{(y+2)^2}{36} = 1$

3. Originally Posted by ScottStedman
I was unfortunately absent on the day of graphing off-center conics in our Algebra 2 class. I've got the notes, but...I could use some help with this.
To learn, in general, how to find an ellipse's equation from given information, try some online lessons.