Question

Find the vertex, focus, directrix and latus rectum of the parabola x² - 4x -  5y -  1 =  0.

Answer 1

The given equation is written as

(x - 2)² = 5(y + 1) = 4(5/4)(y + 1)

Let X = x − 2 and Y = y + 1 so that 

X² = 4aY

where a = 5/4. Therefore

1.  The vertex (X = 0, Y = 0) is (2, -1)

2.  The focus (X = 0,Y = a) is (x = 2, y + 1 = 5/4) = (2, 1/4)

3.  The directrix equation is

Y = -a or y + 1 = -5/4 or 4y + 9 = 0

4.  The latus rectum is 4a = 4(5/4) = 5.