Question

in two or more complete sentences, explain how you would find the equation of a parabola, given the coordinate of the focus and the equation of the directrix.

Explanation:

First, recall the core definition of a parabola: it is the set of all points $(x,y)$ that are equidistant from the given focus $(h,k)$ and the directrix (a line, typically written as $y = c$ for vertical axis parabolas or $x = c$ for horizontal axis parabolas).

To derive the parabola's equation, start by writing the distance formula between a general point $(x,y)$ on the parabola and the focus $(h,k)$, then set this equal to the distance from $(x,y)$ to the directrix. For a horizontal directrix $y = c$, the distance from $(x,y)$ to the directrix is $|y - c|$. Square both sides of the equality to eliminate the square root and absolute value, then simplify the resulting expression to get the standard form of the parabola's equation.

To find the directrix if only the focus is given, you need additional context (like the vertex, which is the midpoint between the focus and directrix for a standard parabola). If the vertex is at $(h,k)$ and the focus is at $(h, k+p)$ (for a vertical parabola opening up/down), the directrix is the line $y = k - p$; for a horizontal parabola with focus $(h+p, k)$, the directrix is $x = h - p$.

Answer:

To find the equation of a parabola from its focus, start with the parabola's definition: all points $(x,y)$ on the parabola are equidistant from the focus and the directrix. Let the focus be $(h,k)$ and the directrix be a line (e.g., $y = c$ for a vertical-axis parabola). Write the distance from $(x,y)$ to the focus $\sqrt{(x-h)^2 + (y-k)^2}$ and set it equal to the distance from $(x,y)$ to the directrix $|y - c|$. Square both sides to eliminate radicals and absolute values: $(x-h)^2 + (y-k)^2 = (y - c)^2$. Simplify this expression by expanding terms and rearranging to get the standard form of the parabola's equation (e.g., $(x-h)^2 = 4p(y - k)$ where $p = k - \frac{c+k}{2}$, the distance from vertex to focus).

To find the directrix, if the parabola is in standard form, use the relationship between the focus and directrix: the vertex is the midpoint between the focus and directrix. For a vertical parabola with focus $(h, k+p)$, the directrix is the horizontal line $y = k - p$; for a horizontal parabola with focus $(h+p, k)$, the directrix is the vertical line $x = h - p$. If only the focus is given, you must know the vertex (or the orientation/scale of the parabola) to calculate the directrix, as the directrix is a line that is symmetric to the focus across the parabola's vertex.