The equation of the parabola in standard form whose focus is (2, 5) and directrix is y = 3 is equal to the quadratic equation y = 0.25 · x² - x + 5. (Correct answer: C)
How to derive the equation of a parabola
Since the directrix is a horizontal line, then we have a parabola with a vertical axis of symmetry. The standard equation of the parabola is described below:
y - k = [1/(4 · p)] · (x - h)² (1)
Where:
- (h, k) - Location of the vertex.
- p - Distance between vertex and focus.
By analytical geometry we understand that the least distance between the focus and the directrix is twice as the distance between the vertex and the focus. First, we find the coordinates of the vertex:
V(x, y) = 0.5 · F(x, y) + 0.5 · D(x, y)
V(x, y) = 0.5 · (2, 5) + 0.5 · (2, 3)
V(x, y) = (1, 2.5) + (1, 1.5)
V(x, y) = (2, 4)
By Pythagorean theorem we find that the distance between vertex and focus is 1. Then, the equation of the parabola in standard form is:
y - 4 = 0.25 · (x - 2)²
y - 4 = 0.25 · (x² - 4 · x + 4)
y - 4 = 0.25 · x² - x + 1
y = 0.25 · x² - x + 5
The equation of the parabola in standard form whose focus is (2, 5) and directrix is y = 3 is equal to the quadratic equation y = 0.25 · x² - x + 5. (Correct answer: C)
To learn more on parabolae: https://brainly.com/question/4074088
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