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I have an ellipse: $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $$ This may be a simple question, but my mind plays tricks on me at the moment;

Which is the most efficient way if I have $x$, $a$ and $b$ and want to find the value of $y$?

Hope someone can help me - thanks in advance :)!

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2 Answers 2

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We can subtract $\dfrac{x^2}{a^2}$ from both sides:


Next multiplying both sides by $b^2$:


Which becomes:


The b's on the left hand side cancel leaving us with:


We can take the square root of both sides leaving us with our final answer:


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You can rearrange $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ to $$y^2 = b^2(\frac{x^2}{a^2} - 1)$$

With knowledge of $a,b$ and $x$ you can evaluate $y^2$ and $$y_{1,2} = \pm \sqrt{y^2} = \pm \sqrt{b^2(\frac{x^2}{a^2} - 1)}$$

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