Find an equation of parabola in the graph below that passes

Find an equation of parabola in the graph below that passes through the given points. The red point represents the vertex. Then, find the focus, directrix, and latus rectum points.

Solution

solution:

equation of parabola with horizontal axis can be written as

x=ay^2+by+c

now we have three (0,-6) , (0,4), (2,-1)

0=36a-6b+c

0=16a+4b+c

2=a-b+c

solving three equations simultaneously

we get the equation

x= -2y^2/25 -4y/25 +48/25

focus is (-9/8,-1)

directrix is x=41/8

lengthof latus rectum=25/2