Question

Find the equation of a parabola with vertex at $\left(-1,2\right)$and focus at$\left(-1,3\right)$.

Short answer

The equation of parabola is${\left(x+1\right)}^2=-12\left(y-2\right)$

Step-by-step solution

Step 1. Given Information

Vertex of the parabola $\left(-1,2\right)$and focus $\left(-1,3\right)$.

We need to find the equation of the parabola.

Step 2. Equation of Parabola

Here

$$\begin{gathered} k+a=-1 \\ 2+a=-1 \\ a=-3 \end{gathered}$$

Thus, equation of parabola is

$$\begin{gathered} {\left(x-h\right)}^2=4a\left(y-k\right) \\ {\left(x+1\right)}^2=4\left(-3\right)\left(y-2\right) \\ {\left(x+1\right)}^2=-12\left(y-2\right) \end{gathered}$$