To derive the equation of the parabola, let (x , y) be a point in the parabola. Its distance from the focus should be equal to its distance from the directrix.
I'll show here how to do number 1.
1.) focus : (4, -7) and directrix of y = -15
d (point to focus) = d (point to directrix)
sqrt ((x - 4)² + (y + 7)²) = (y + 15)
Squaring both sides gives us,
(x - 4)² + (y + 7)² = (y + 15)²
Simplifying gives,
x² - 8x + 16 + y² + 14y + 49 = y² + 30y + 225
Simplifying leads to,
16y = x² - 8x -160
This leads to our final answer of
y = (1/16)x² - (1/2)x - 10
All the rest of the numbers follow the same steps to get answered.
