It is difficult for me to learn that **ellipse**, I use **The elements of the coordinate geometry** by **S.L. alone** and his derivation of the equation into an ellipse goes like this and , As you can see, when he says so *Let C be the origin* and that simplifies everything and because this is just that we get a nice shape like this $$ frac {x ^ 2} {a ^ 2} + frac {y ^ 2} {b ^ 2} = 1 $$ I want to know how to find the equation for an ellipse in the center $ C $ should be with $ (h, k) $,

Here is my attempt: look at the coordinates of $ C $ his $ (h, k) $ and the coordinates of $ S $ his $ (p, q) $, since $ CS $ is $ a ~ e $, that's why $$ (p-h) ^ 2 + (q-k) ^ 2 = a ^ 2 ~ e ^ 2 $$

and similar coordinates of $ Z $ can be accepted $ (c, d) $ and again $$ (c-h) ^ 2 + (d-k) ^ 2 = 1 $$, Let us now turn to our example point $ P (x, y) $, by defining the ellipse: $$ SP ^ 2 = e ^ 2 PM ^ 2 $$

$$ (p-x) ^ 2 + (q-y) ^ 2 = e ^ 2 NZ ^ 2 $$ and now the problem comes, I can not write $ N $ how $ (x, 0) $ there $ ZCZ & # 39; $ is no longer that $ x- $Axis. Why only extracts that we get this problem? It was so easy with circles and parabolas, but when I became acquainted with Ellipse I had problems. You, the equation of the parabola $ y ^ 2 = 4ax $ Tell us that the opening is positive $ x- $Axis and it lies on $ x- $Axis and $ x ^ 2 = 4ay $ just tell us everything is on now $ y- $ Axis, but in the case of ellipses, we have to determine by comparison what the eclipse would look like $ a $ and $ b $ (half the major and minor axis lengths). If we are given an equation like $$ frac {x ^ 2} {a ^ 2} + frac {y ^ 2} {b ^ 2} = 1 $$ and said that the latus rectum would then find which formula we should use $ frac {2b ^ 2} {a} $ or $ frac {2a ^ 2} {b} $ (If it comes into question, it will only be given $ a $ and $ b $ and not said which is bigger).

Finally, I apologize for posting the pictures and not the *thing written in latex*, Well, I did it so that nothing should be missing, as there is a possibility of looking at something less worth mentioning when writing, but in reality it could be worthy. So literally the best thing is to avoid misunderstandings.

Many thanks.