What is the equation of the quadratic graph with a focus of (3, 6) and a directrix of y=4?
Solution:
Let P(x, y) be the moving point.
It is the locus of a point P which moves such that the distance of the point from focus S(3, 6) and the directrix y = k = 4 is equal.
Draw PM perpendicular to y = k = 4, then coordinates of M(x , 4)
By definition and the diagram,
PS = PM
Squaring both the sides,
PS2 = PM2
(x - 3)2+ (y - 6)2= (x - x)2+ (y - 4)2
x2 -6x + 9 + y2 - 12y + 36 = y2 -8y + 16
x2 -6x + 9 - 12y + 36 = -8y + 16
x2 -6x + 9 = 4y - 20
(x - 3)2= 4(y - 5), which is of the form (x - h)2= 4a (y - k).
What is the equation of the quadratic graph with a focus of (3, 6) and a directrix of y=4?
Summary:
The equation of the quadratic graph with a focus of (3, 6) and a directrix of y=4 is (x-3)2= 4(y - 5)