Question

A circle who's center is (4,-1) passes through focus of parabola x[sup]2[/sup]+16y=0 show that circle touches directrix of parabola.

Answer 1

Represent equation of parabola in the form: x² = 2px:
X² = 2 (-8) y
whence p = -8.

Then the focus has coordinate (0, p/2) = (0,-4)
And the directrix has equation:
Y = -p/2=4

Then the line segment from the center (4,-1) of a circle to the focus (0,-4) is equal to

√(4² + (-1+4)²)=5

The directrix is a horizontal line y=4, and its distance to the center (4,-1) is equal
4-(-1) = 5,
which coincides with the radius.
Hence the circle touches the directrix.