What is the equation of the quadratic graph with a focus of (4, 3) and a directrix of y = -6?
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 which moves such that the distance of the point from focus and the directrix is equal. Here it is given that the focus is S(4, 3) and the directrix is y = k = -6. Draw PM perpendicular to y = k = -6 then, coordinates of M(x, -6).
By definition and the diagram,
PS = PM
Squaring both the sides,
PS2 = PM2
(x - 4)2 + (y - 3)2= (x - x)2 + (y + 6)2(using the distance formula between two points)
x2 - 8x + 16 + y2 - 6y + 9 = y2 + 12y + 36
x2 - 8x + 16 - 6y + 9 = 12y + 36
x2 - 8x + 16 = 18y + 27
(x - 4)2 = 18y + 27
(x - 4)2 = 18[y + (3/2)], which is of the form (x - h)2 = 4a( y - k).
What is the equation of the quadratic graph with a focus of (4, 3) and a directrix of y = -6?
Summary:
The equation of the quadratic graph with a focus of (4, 3) and a directrix of y = -6 is (x - 4)2 = 18[y + (3/2)].