A parabola can be drawn given a focus of left bracket, 5, comma, 6, right bracket(5,6) and a directrix of y, equals, minus, 10y=−10. What can be said about the parabola?

Answer 1

The parabola opens away from the directrix with the focus being in the interior of the parabola. That means the parabola opens upward. Since the focus lies on the axis of symmetry, the equation for the axis of symmetry is x = 5. The points (5,6) and (5,-10) lie on the axis of symmetry and the vertex is the midpoint of the segment joining those points, so (5,-2).

At this point we have y = 1/(4p) (x - 5)² - 2, where p is the directed distance from the focus to the vertex, i.e. 6 - (-2) = 8.

So the equation of the parabola is y = 1/32 (x -5)² - 2. Any point on the parabola is the same distance from the focus as it is from the directrix.

Use the slider for the point on the parabola and monitor the distance from P to Q and from P to F.

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