A parabola has a focus (3,-2) and a directrix y=2. The point (a,-8) is on the parabola. How far is this point from the focus?
To find the distance between the point (a,-8) and the focus (3,-2), we need to use the formula for finding the distance between a point (x1, y1) and a point (x2, y2):
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the coordinates of the point (a,-8) are (a,-8) and the coordinates of the focus (3,-2) are (3,-2).
So, the distance d between the point (a,-8) and the focus (3,-2) is:
d = sqrt((3 - a)^2 + (-2 - (-8))^2)
d = sqrt((3 - a)^2 + 6^2)
d = sqrt((3 - a)^2 + 36)
d = sqrt(9 - 6a + a^2 + 36)
d = sqrt(a^2 - 6a + 45)
Since the point (a,-8) lies on the parabola, it satisfies the definition of a parabola: the distance from the point to the focus is equal to the distance from the point to the directrix.
So, we need to find the perpendicular distance from the point (a,-8) to the directrix y=2.
The perpendicular distance between a point (x1, y1) and a line Ax + By + C = 0 is given by the formula:
d = |Ax1 + By1 + C| / sqrt(A^2 + B^2)
In this case, the equation of the directrix is y=2. So, we have A=0, B=1, and C=-2.
Therefore, the perpendicular distance from the point (a,-8) to the directrix y=2 is:
d = |0*a + 1*(-8) + (-2)| / sqrt(0 + 1)
d = | -8 - 2 | / sqrt(1)
d = 10 / 1
d = 10
Since the point (a,-8) lies on the parabola, the distance between the point and the focus is equal to the distance between the point and the directrix.
So, the distance between the point (a,-8) and the focus (3,-2) is 10 units.