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Chapter 6: Problem 56

Find the standard form of the equation of the parabola with the given characteristics. Focus: \((0,0) ;\) directrix: \(y=8\)

Short Answer

Expert verified
The standard form of the equation of the parabola is \(y= -1/16x^2 + 4\)

Step by step solution

01

Find the Vertex

The vertex is halfway between the focus and the directrix. Since the focus is at (0,0), and the directrix is \(y=8\), the vertex of the parabola is at \(0, h\), where \(h\) is the y coordinate of the vertex and it equals \((8 - 0)/2 = 4\). So the vertex is at (0,4).
02

Determine the Orientation of the Parabola

Since the focus of the parabola is below the directrix, the parabola opens downward. This influences the sign of the \(a\) in the standard form of the parabola, which will be negative.
03

Find the 'a' Value

The value of \(a\) is \(1/(4p)\), where \(p\) is the distance from the vertex (0,4) to the focus (0,0) or the directrix \(y=8\). The value of \(p\) is therefore 4, so \(a=1/(4*4)=-1/16\), taking into account the negative sign because the parabola opens downward.
04

Write the Equation in Standard Form

The standard form of the equation of the parabola is \(y=a(x - h)^2 + k\). Putting the known values, the equation is: \(y= -1/16(x - 0)^2 + 4\) simplifying this we get: \(y= -1/16x^2 + 4\)

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