What is the equation of the quadratic graph with a focus of (4,0) and a directrix of y=10?

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 focus S(4, 0) and the directrix y= k = 10. Draw PM perpendicular to y = k = 10, then coordinates of M(x , 10)

By definition and the diagram,

PS = PM

Squaring both the sides,

PS² = PM²

(x-4)²+ (y-0)²= (x-x)²+ (y-10)²(using the distance formula between two points)

x² -8x + 16 + y² = y² -20y + 100

x² -8x + 16 = -20y + 100

x² -8x + 16 = -20(y - 5)

(x-4)²= -20(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 (4,0) and a directrix of y=10?

Summary:

The equation of the quadratic graph with a focus of (4,0) and a directrix of y=10 is (x-4)²= -20(y - 5).