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What is the equation of the quadratic graph with a focus of (2, 0) and a directrix of y = -12?
Solution:
Let P(x, y) be the moving point. A quadratic graph is that of a parabola. The parabola is the locus of a point P that moves such that the distance of the point from focus and the directrix is equal. Here focus S(2, 0) and the directrix y = -12. Draw PM perpendicular to y = k = -12 then, coordinates of M(x, -12).
By definition and the diagram,
PS = PM
Squaring both the sides,
PS2 = PM2
(x - 2)2 + (y - 0)2= (x - x)2 + (y + 12)2 (using the distance formula between two points)
x2 - 4x + 4 + y2 = y2 + 24y + 144
x2 - 4x + 4 = 24y + 144
(x - 2)2 = 24(y + 6), which is of the form (x - h)2 = 4a(y - k).
What is the equation of the quadratic graph with a focus of (2, 0) and a directrix of y = -12?
Summary:
The equation of the quadratic graph with a focus of (2, 0) and a directrix of y = -12 is (x - 2)2 = 24(y + 6).
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