How to Find the Directrix & Focus in a Parabola

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The focus and directrix are a point and line that define a parabola. The focus, which lies inside the parabola, is always the same distance from a specific point on the surface of the parabola as that point is from the directrix. The directrix is outside of the parabola and forms a 90-degree angle with its axis of symmetry. Finding the focus and directrix in a parabola is a moderately easy task that can be completed using the equation of a given parabola.

  • Rearrange the equation of the parabola. The vertex form of the parabola equation is y = a(x - h)^2 + k. As an example, consider the parabola y = 2(x - 4)^2 + 6 as an example. The form you need the equation to be in is (x - h)^2 = 4p(y - k). You can rearrange the equation by subtracting "k" from "y" and dividing both sides of the equation by "a." The example is rearranged as (x - 4)^2 =1/2(y - 6). After doing this, you should note that 4p = 1/a. In the example, 4p=1/2

  • Calculate the y-value for the parabola's focus. The x-value is already known to be "h" in the parabola's equation. The y-value now needs to be found. It can be found by solving for "p" in the equation 4p = 1/a. After dividing both sides by four, you will find that p = 1/4a or 1/8 in the example. The y-value for the parabola's focus is k + p. The focus of the parabola is therefore the point located at (h, k + p) or (4, 49/8).

  • Find the parabola's directrix. The directrix is opposite the focus. It, therefore, has the equation y = k - p or, in the example, y = 47/8.

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