How do you write an equation for the parabola with Vertex at the origin, focus at ( 0, 1/11)?

Jul 18, 2016

$\textcolor{red}{11 {x}^{2} = 4 y}$

Explanation:

Given that the vertex of parabola is origin $\left(\textcolor{red}{0 , 0}\right)$ and focus $\left(\textcolor{red}{0 , \frac{1}{11}}\right)$.So the directrix of the parabola will be perpendicular to y-axis and will pass through $\left(\textcolor{red}{0 , - \frac{1}{11}}\right)$.Hence its equation will be $\textcolor{b l u e}{y = - \frac{1}{11}}$.

Let $\left(\textcolor{g r e e n}{h , k}\right)$ be the coordinate of any point on the parabola.

So by definition of parabola

$\text{Dist.of(h,k) from focus"="Dist. of(h,k) from directrix}$

$\sqrt{{h}^{2} + {\left(k - \frac{1}{11}\right)}^{2}} = \left(k + \frac{1}{11}\right)$

$\implies \left({h}^{2} + {\left(k - \frac{1}{11}\right)}^{2}\right) = {\left(k + \frac{1}{11}\right)}^{2}$

$\implies {h}^{2} = {\left(k + \frac{1}{11}\right)}^{2} - {\left(k - \frac{1}{11}\right)}^{2}$

$\implies {h}^{2} = 4 \cdot k \cdot \frac{1}{11}$

$\implies 11 {h}^{2} = 4 k$

Substituting x for and y for k we get the equation of the parabola as follows.

Equation of parabola

$\textcolor{red}{11 {x}^{2} = 4 y}$