Final answer:
To determine the equation of the parabola with focus (1,5) and directrix y=-15, we can use the standard form of a parabola equation: (x-h)^2=4p(y-k), where (h,k) is the vertex and p is the distance between the vertex and the focus or directrix.
Step-by-step explanation:
To determine the equation of the parabola with focus (1,5) and directrix y=-15, we can use the standard form of a parabola equation: (x-h)^2=4p(y-k), where (h,k) is the vertex and p is the distance between the vertex and the focus or directrix. In this case, the vertex is (h,k) = (1, 5) and the directrix is y = -15.
Therefore, we have (x-1)^2 = 4p(y-5).
To find the value of p, we can use the distance formula between the vertex and the directrix. The distance between the vertex (1,5) and the directrix y = -15 is 20 units.
Since p represents the distance between the vertex and the focus or directrix, we have 4p = 20, which gives us p = 5.
Now we can substitute the values into the equation: (x-1)^2 = 4(5)(y-5). Simplifying, we get (x-1)^2 = 20(y-5). This is the equation of the parabola.