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Chapter 6: Problem 44

Find the vertex, focus, and directrix of the parabola. Use a graphing utility to graph the parabola. $$x^{2}-2 x+8 y+9=0$$

Short Answer

Expert verified
The vertex of the parabola is (1, -1.125), the focus is at (1, -3.125), and the equation of the directrix is \(y = 0.875\)

Step by step solution

01

Convert the equation into standard form

Rearrange the given equation to get \(x^{2}-2 x = -8 y-9\), and complete the square on the left side to convert this into the standard form: \((x-1)^2 = -8(y+1.125)\)
02

Identify the Vertex

The vertex in the standard form is \((h, k)\), so the vertex in the equation \((x-1)^2 = -8(y+1.125)\) is \((1, -1.125)\)
03

Find the value of \(a\)

According to the standard form of a parabola \((x-h)^2 = 4a(y-k)\): \(4a = C\), hence \(a = C/4 = -8/4 = -2\)
04

Find the Focus

The focus of the parabola is located at \((h, k + a)\), substituting the obtained values, the focus is at \((1, -1.125 - 2) = (1, -3.125)\)
05

Find the Directrix

The directrix of a parabola is \(\; y = k - a\). Substituting the values of \(k\) and \(a\) found, the directrix is \(y = -1.125 + 2 = 0.875\)

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