Finding the Equation of a Parabola Given Focus and Directrix
Given thefocusanddirectrixof aparabola, how do we find the equation of the parabola?
If we consider only parabolas that open upwards or downwards, then the directrix will be ahorizontal lineof the form.
Letbe the focus and letbe the directrix. Letbe any point on the parabola.
一个ny point,on the parabola satisfies the definition of parabola, so there are two distances to calculate:
- Distance between the point on the parabola to the focus
- Distance between the point on the parabola to the directrix
To find the equation of the parabola, equate these two expressions and solve for.
Find the equation of the parabola in the example above.
Distance between the pointand:
Distance between pointand the line:
(Here, the distance between the point and horizontal line is difference of their-coordinates.)
Equate the two expressions.
Square both sides.
Expand the expression inon both sides and simplify.
This equation inis true for all other values on the parabola and hence we can rewrite with.
Therefore, the equation of the parabola with focusand directrixis
Example:
If the focus of a parabola isand the directrix is, find the equation of the parabola.
Letbe any point on the parabola. Find the distance betweenand the focus. Then find the distance betweenand directrix. Equate these two distance equations and the simplified equation inandis equation of the parabola.
The distance betweenandis
The distance betweenand the directrix,is
.
把两个距离表达式和广场both sides.
Simplify and bring all terms to one side:
Write the equation withon one side:
This equation inis true for all other values on the parabola and hence we can rewrite with.
So, the equation of the parabola with focusand directrix isis