Calculating a Point's X position on an Ellipse, given pos Y

Question

How can I calculate for the X coordinate given the Y value of the position? Which the Y position is 10 units, as seen in the image below.

We know the X diameter is 200 and the Y diameter is 150.

Answer 1

The ellipse equation here is $$ \frac{x^2}{100^2} + \frac{y^2}{75^2} = 1 $$ so $$ x = \pm\sqrt{100^2\left(1-\frac{y^2}{75^2}\right)} = \pm 100 \sqrt{1-\frac{y^2}{75^2}} $$ Since you want a positive value (from your picture), pick the "+" in the $\pm$.

Answer 2

If you have an ellipse, you can use the standard equation for an ellipse:

$$\left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 = 1,$$

where $a = \frac{200}{2}$ and $b = \frac{150}{2}$. Then you can find the $x$ value simply by substituting your $y = 10$ value and solving the above equation algebraically for $x$.

Note that when you solve for $x$, you will have to take a square root, which means you should have a $\pm$ in your answer (since there are two x-values for which $y = 10$).

You can read more information about ellipses here:

https://en.wikipedia.org/wiki/Ellipse#Equations