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Chapter 6: Problem 55

Find the standard form of the equation of the parabola with the given characteristics. Focus: \((2,2) ;\) directrix: \(x=-2\)

Short Answer

Expert verified
The standard form of the equation of the parabola is \(x=0.125(y-2)^2\).

Step by step solution

01

Determine the Direction of the Parabola

In the given problem, the focus of the parabola is at point \((2,2)\) and the directrix of the parabola is at \(x=-2\). Since the x-coordinate of the focus is greater than the x-coordinate of any point on the directrix, we can conclude that the parabola opens to the right.
02

Calculate the Vertex

The vertex of a parabola is the midpoint of the line segment joining the focus and any point on the directrix. The point on the directrix vertically aligned with the focus is \((-2,2)\). So the vertex, being the midpoint of \((2,2)\) and \((-2,2)\), is \((0,2)\).
03

Formulate the Equation of the Parabola

The general form of the equation for a right-opening parabola is \(x=a(y-k)^2+h\), where \((h,k)\) is the vertex and \(a\) is \(\frac{1}{4c}\), where c is the distance from the vertex to the focus or directrix. The distance from vertex \((0,2)\) to focus \((2,2)\) is 2 units, hence \(a=\frac{1}{4*2}=0.125\). Substituting these values, the equation of the parabola becomes \(x=0.125(y-2)^2+0\).

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