I am given the following problem:
Suppose that P is a planet that describes an elliptical orbit around the star O located at ( 0,0 ). Based on the image below, find the distance from P to O.
What I have so far is the distance between the focus points and O:
\begin{align*} a &= 10\\ b &= 5\\ 100 &= 25+c^2 \therefore c = \sqrt{75} \end{align*}
I am not sure how to proceed now. I would appreciate some help, thanks.
Start by forming a right triangle as shown.
$\hskip{.75in}$
From here, remember that
$$\tan(\alpha) = \frac{x}{y}.$$
Since $\alpha = 45^\circ$, and $\tan(45^\circ) = 1$, this leads us to:
$$ 1 = \frac{x}{y} \quad \Rightarrow \quad y=x.$$
Since you found $a = 10$, $b=5$ and we are given the center is $(0,0)$, we then know this ellipse has the standard form
$$ \frac{x^2}{100} + \frac{y^2}{25} = 1.$$
Since $y=x$, this turns into: $$ \frac{x^2}{100} + \frac{x^2}{25} = 1.$$
Solving for $x$ in this equation will give you $P$. Then use the distance formula to find what the question is asking.
Can you take it from here?