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: x^{2} = 2px:

X^{2} = 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

√(4^{2} + (-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.

X

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

√(4

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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