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Chapter 6: Problem 59

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 equation of the parabola is \(y^2 - 4y + 4 = 8x\).

Step by step solution

01

Find the Vertex of the Parabola

The vertex of a parabola is the midpoint between the focus and the directrix. In this case, given a focus at (2, 2) and a directrix at \(x=-2\), the vertex occurs at \((h, k)\) where \(h = \frac{(2 + -2)}{2} = 0\) and \(k \) is the y-coordinate of the focus which is 2. Therefore, the vertex is at the point (0, 2).
02

Formulate the Equation of the Parabola

The equation of a parabola can be given in the form \((x-h)^2 = 4p(y-k)\) if the parabola opens upward or downward. Since the directrix is \(x = -2\) (a vertical line) and the focus is to the right of the directrix, this parabola opens to the right. Hence, we use the form \((y-k)^2 = 4p(x-h)\). Here, \(p\) is the distance from the vertex to the focus or directrix. In this case, \(p = 2\). Hence, substituting into the equation, we get \((y-2)^2 = 4(2)(x-0)\).
03

Simplify the Equation

Expanding and arranging the equation \((y-2)^2 = 8(x-0)\), we get the equation \(y^2 - 4y + 4 = 8x\). This can be rewritten as \(y^2 - 4y = 8x - 4\). Adding \(-4\) on both sides, we get \(y^2 - 4y + 4 = 8x\). Therefore, the standard form of the equation of the parabola is \(y^2 - 4y + 4 = 8x\).

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. It is possible for a parabola to intersect its directrix.

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