What is the equation of the quadratic graph with a focus of (5, -1) and a directrix of y = 1?

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(5, -1) and the directrix y = k= 1. Draw PM perpendicular to y = k = 1 then, coordinates of M(x, 1).

By definition and the diagram,

PS = PM

Squaring both the sides,

PS² = PM²

(x-5)²+ (y+1)²= (x-x)²+ (y-1)²(using the distance formula between two points)

x² -10x + 25 + y² +2y + 1 = y² -2y + 1

x² -10x + 25 = -4y

(x-5)² = -4(y-0), which is of the form (x- h)²= 4a (y - k).

What is the equation of the quadratic graph with a focus of (5, -1) and a directrix of y = 1?

Summary:

The equation of the quadratic graph with a focus of (5,-1) and a directrix of y=1 is (x-5)² = -4(y-0).