Final answer:
The standard form of the parabola with focus (3, 8) and directrix x = -7 is x = -(y - 3)^2 + 8, which corresponds to option d).
Step-by-step explanation:
To find the equation in standard form of the parabola with focus (3, 8) and directrix x = -7, we need to use the formula for the equation of a parabola. The standard form of a parabola equation is (x - h)² = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix. In this case, the vertex is (3, 8) and the directrix is x = -7.
The distance from the vertex to the directrix is 3 - (-7) = 10, so p = 10. Plugging the values into the equation, we get (x - 3)² = 4(10)(y - 8). Simplifying, we get (x - 3)² = 40(y - 8). Therefore, the equation in standard form of the parabola is (x - 3)² = 40(y - 8).
The student is seeking the equation in standard form of a parabola with a given focus (3, 8) and a directrix x = -7. To find the standard form of the parabola, we use the definition that a parabola is the set of all points (x, y) that are equidistant from the focus and the directrix. The distance from a point (x, y) to the focus (3, 8) is the same as the distance from (x, y) to a point (x', -7) where x' is the x-coordinate of the point (x, y).
To express this equality of distances, we use the distance formula:
sqrt((x - 3)^2 + (y - 8)^2) = sqrt((x - x')^2 + (y - (-7))^2). Since x = x', this simplifies to y - 8 = -(x - 3)^2 since the parabola opens to the left (towards the negative x-axis). Thus, the standard form equation of the parabola is x = -(y - 3)^2 + 8. This corresponds to option d), which is the correct answer.