Question

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (- 5, 0); Directrix: x = 5

Accepted Answer

Recall that the standard form of a parabola depends on its orientation. Since the focus and directrix are vertical lines (focus at (-5, 0) and directrix at x = 5), the parabola opens horizontally. Find the vertex of the parabola, which lies exactly halfway between the focus and the directrix. Calculate the midpoint of the x-coordinates of the focus and directrix: $$x_v = \frac{-5 + 5}{2} = 0. $$The y-coordinate of the vertex is the same as the focus's y-coordinate, so $$y_v = 0. $$Thus, the vertex is at $$(0, 0). $$Determine the distance $$p$$ from the vertex to the focus (or directrix). Since the focus is at $$(-5, 0)$$ and the vertex at $$(0, 0)$$, $$p = -5 ($$negative because the parabola opens to the left). Use the standard form of a horizontally opening parabola with vertex at $$(h, k)$$: $$(y - k)^2 = 4p(x - h). $$Substitute $$h = 0$$, $$k = 0$$, and $$p = -5 to $$get $$y^2 = 4(-5)(x - 0$$)$$. Simplify the equation to get the standard form: y^2 = -20x$. This is the equation of the parabola with focus (-5$$, 0)$$ and directrix x = 5$.

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (- 5, 0); Directrix: x = 5

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Key Concepts

Definition of a Parabola

Standard Form of a Parabola

Finding the Vertex from Focus and Directrix