What if the directrix is not parallel to an axis? Suppose it is y=x+2, and let the focus be ((4,0)
How do I derive the equation of the parabola? The distance from the parabola is perpendicular to the directrix.
1. Use the complete definition of the parabola:
The distance of an arbitrary point P of the parabola to the focus F is equal to the perpendicular distance to the directrix d (= blue lines)
2. The triangle QFP is isosceles and the line MP is the perpendicular bisector of QF (by the way: MP is simultaneously the tangent to the parabola)
3. As posted the directrix has the equation y = x+2 and the focus is F(4,0). The point Q is located on the directrix and has the coordinates Q(q, q+2)
4. Then M has the coordinates
5. The slope of QF is and therefore the perpendicular slope is
6. The equation of the perpendicular bisector of QF is:
which will yield:
7. The perpendicular line to directrix passing through Q has the equation:
8. Now calculate the coordinates of the point P which is the intersection between the lines of #6 and #7:
After a lot of transformation you'll get:
9. Plug in this value into the equation at #7 to calculate the y-coordinate of P:
10. Now you have a parametric equation of the parabola:
11. Solve for q and plug this term into which will yield:
After Jen and I saw Earboths amazing post, she said to me can't you do that with a matrix rotation. I thought yes, you should be able to. So Here it is.
The directrix can be written as a vector
and the focus as .
Since the directrix has slope 1 I will rotate it 45 degrees clockwise.
Note that the inverse of R is the transpose of R.
Now we can transform the focus and the directrix
Now in the rotated system the equation of the directrix is
Now the form of the parabola is
We can identify h and k as
p is the vertical distance from the vertex to the focus
So we get the equation
Solving this for y gives
Now we can write this in vector form
Now to rotate back we use
Now we have parametric equation for the parabola
Now we can eliminate the parameter
So we end up with
This turns out to be the same as Earboth's if you move some stuff around.