Given any point on the parabola, the distance from that point to the focus is equal to the closest point on the directrix. Then, by using the distance formula, the lengths can be compared to get the equation for the parabola.
1
Draw line segments
expand_more
Any point on a parabola has the same distance to the focus, F, as to the closest point, D, on the directrix. Thus, line segments are drawn from an arbitrary point, P, on the parabola, to F and D.
2
Use the distance formula
expand_more
The length of each segment can be expressed with the distance formula since it's the distance between two points.
The coordinates for the focus can be read from the coordinate plane. Since the arbitrary point is any point on the parabola, its coordinates are unknown. Thus, the point D, on the directrix, shares the same x-value as P and has the y-value 1.
F(-2,-5)P(x,y)D(x,1)
The points can now be substituted into the expressions for the lengths.
Mathleaks uses cookies for an enhanced user experience. By using our website, you agree to the usage of cookies as described in our policy for cookies.