Answer to Question #2567 in Analytic Geometry for Harshad D Kohale

Question #2567
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.
Expert's answer
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.

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