The equation of the parabola in standard form whose focus is (2, 5) and directrix is y = 3 is equal to the <em>quadratic</em> equation y = 0.25 · x² - x + 5. (Correct answer: C)
<h3>How to derive the equation of a parabola</h3>
Since the directrix is a <em>horizontal</em> line, then we have a parabola with a <em>vertical</em> 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 <em>least</em> 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 <em>standard</em> 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 <em>quadratic</em> equation y = 0.25 · x² - x + 5. (Correct answer: C)
To learn more on parabolae: brainly.com/question/4074088
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