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# The focus of a parabola is located at (4,0) and the directrix is located at x = -4. Which equation represents the parabola?

y^{2} = -x

y^{2} = x

y^{2} = -16x

y^{2} = 16x

**Solution:**

The definition of a parabola states that all points on the parabola always have the same distance to the focus and the directrix.

Let A = (x,y) be a point on the parabola.

Focus, F = (4,0)

Given, directrix x = -4

D = (-4, y) represent the closest point on the directrix

First, find out the distance using distance formula,

d = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Distance A and F is dAF = √(x - 4)^{2} + (y - 0)^{2}

Distance between A and D is dAD = √(x - (-4))^{2} + (y - y)^{2} = √(x + 4)^{2}

Since these distances must be equal to each other,

√(x - 4)^{2} + (y - 0)^{2} = √(x + 4)^{2}

Squaring both sides,

(√(x - 4)^{2} + (y - 0)^{2})^{2} = (√(x + 4)^{2})^{2}

(x - 4)^{2} + y^{2} = (x + 4)^{2}

x^{2} - 8x + 16 + y^{2} = x^{2} + 8x + 16

y^{2} = 8x + 8x

y^{2} = 16x

Therefore, the equation of the parabola is y^{2} = 16x

## The focus of a parabola is located at (4,0) and the directrix is located at x = -4. Which equation represents the parabola?

**Summary:**

The focus of a parabola is located at (4,0) and the directrix is located at x = -4. The equation of the parabola is y^{2} = 16x.

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