# 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,

PS^{2} = PM^{2}

(x - 2)^{2 }+ (y - 0)^{2}= (x - x)^{2 }+ (y + 12)^{2 }(using the distance formula between two points)

x^{2} - 4x + 4 + y^{2} = y^{2} + 24y + 144

x^{2} - 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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