Question

write an equation of the parabola with the given characteristics

Answer 1

Given:

The focus of parabola is (-3,11/2).

The equation of directrix is y = -3/2.

Explanation:

The general equation of parabola,

$y-k=\frac{1}{4p}(x-h)^2$

Then coordinates of focus is (h,k+p) and directrix equation is y = k - p.

On comparison with given focus and directrix equation,

$h=-3$$k+p=\frac{11}{2}$$k-p=-\frac{3}{2}$

Add equation k + p = 11/2 and k - p = -3/2 to obtain the value of k.

$\begin{gathered} k+p+k-p=\frac{11}{2}-\frac{3}{2} \\ 2k=4 \\ k=2 \end{gathered}$

Determine the value of p.

$\begin{gathered} 2-p=-\frac{3}{2} \\ p=2+\frac{3}{2} \\ =\frac{7}{2} \end{gathered}$

So value of h is -3, k is 2 and p is 7/2.

Determine the parabola equation for these h, p and k values.

$\begin{gathered} y-2=\frac{1}{4\cdot\frac{7}{2}}(x-(-3))^2 \\ y=\frac{1}{14}(x+3)^2+2 \end{gathered}$

So equation of parabola is,

$y=\frac{1}{14}(x+3)^2+2$