A conic with equation $$ a x^2 + b y^2 = c $$ has two focus points, where $a=4$, $b=24$ and $c=65$. One of those focus points has a positive x-coordinate. To two decimal places, what is the value of that positive x-coordinate?
I got the answer as 14.83, seems a bit too big is that right?
The above answer is according to this in my textbook
Put the ellipse in standard form. Start from $4x^2+24y^2=65$. Divide through by $65$. We get $\frac{4}{65}x^2+\frac{24}{65}y^2=1$.
This is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a^2=\frac{65}{4}$ and $b^2=\frac{65}{24}$. Finally, you want $\sqrt{a^2-b^2}$.
Your intuition is right: this is substantially smaller than your answer. To two decimal places, I get $3.68$.