Write the standard form of the equation of the parabola whose directrix is x=-1 and whose focus is at (5,-2)?
a) (y+2)^2 = 12(x+2)
b) y-2 = 12(x+2)
c) x+2 = 1/12(y+2)^2
d) x-2 = 1/12(y+2)^2
I get A and D? confused!
To find the standard form of the equation of the parabola given the directrix and the focus, you can use the definition of a parabola in terms of its directrix and focus.
The standard form of the equation of a parabola with a vertical axis of symmetry is given by (y-k)^2 = 4a(x-h), where (h,k) represents the vertex and a represents the distance from the vertex to the focus or directrix.
In this case, the directrix is x = -1 and the focus is at (5,-2).
Step 1: Determine the x-coordinate of the vertex, which is the average of the x-coordinates of the focus and directrix.
x-coordinate of the vertex = (x-coordinate of focus + x-coordinate of directrix) / 2
x-coordinate of vertex = (5 + (-1)) / 2 = 4/2 = 2
Step 2: Determine the y-coordinate of the vertex. In this case, the y-coordinate of the vertex is the same as the y-coordinate of the focus, so it is -2.
Step 3: Determine the distance "a" from the vertex to the focus or directrix.
The distance from the vertex to the directrix is the absolute value of the difference between the x-coordinates of the vertex and the directrix.
a = |x-coordinate of vertex - x-coordinate of directrix|
a = |2 - (-1)| = |3| = 3
Step 4: Substitute the values of the vertex (h,k) and the distance "a" into the standard form equation.
(y+2)^2 = 4a(x-2)
Plugging in the values: (y+2)^2 = 4(3)(x-2)
(y+2)^2 = 12(x-2)
This matches option (a): (y+2)^2 = 12(x+2).
So, the correct answer is option (a): (y+2)^2 = 12(x+2).