Question

What are the vertex, focus and directrix of y=(x−1)2−1?

Answer 1

Since given parabola is after arranging to standard form:

(x-1)² = (y+1)

Hence the vertex is at (1,-1)

Axis of parabola is parallel to y axis

The length of latus rectum is 1, therefore:

4a=1 => a=1/4

Now for focus:

Focus is always along the axis of parabola which is parallel to y axis, so the abscissa(x component) must be '1', for ordinate(y component):

y = -1+1/4 => -3/4

Focus = (-1,-3/4)

For directrix:

Directrix must be parallel to x axis, and at a distance of -1/4 further down the vertex

x = -1-1/4

x = -5/4

4x+5=0 --- equation of directrix