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Problem 758

Consider the equation of a parabola $\mathrm{x}^{2}-4 \mathrm{x}-4 \mathrm{y}+8=0$. Find the focus, vertex, axis of symmetry, and the directrix.

Short Answer

Expert verified
The properties of the given parabola are: Vertex: \((2, 1)\), Axis of symmetry: \(x = 2\), Focus: \((\frac{9}{4}, 1)\), and Directrix: \(x = \frac{7}{4}\).
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Step 1: Rewrite the equation in vertex form

First, rewrite the given equation, \(\mathrm{x}^{2}-4 \mathrm{x}-4\mathrm{y}+8=0\), by rearranging the terms and completing the square. 1. Rearrange: \(x^2 - 4x = 4(y - 2)\) 2. Complete the square: \((x - 2)^2 = 4(y - 2) + 4\), which implies that \((x - 2)^2 = 4(y - 2 + 1)\) 3. Simplify: \((x - 2)^2 = 4(y - 1)\) Now, the equation of the parabola is in vertex form: \(x = a(y - k)^2 + h\) with \(a = 1\), \(h = 2\), and \(k = 1\).

Step 2: Find the vertex

The vertex of the parabola is given by the point \((h, k)\). From the vertex form of the equation, we already found that \(h = 2\) and \(k = 1\). So the vertex is \((2, 1)\).

Step 3: Determine the axis of symmetry

The axis of symmetry is a vertical line that goes through the vertex and is parallel to the x-axis. Since the vertex is \((2, 1)\), the equation of the axis of symmetry is \(x = 2\).

Step 4: Calculate the focus

Since the equation is given in the form \(x = a(y-k)^2 + h\) and \(a = 1\), the distance between the vertex and the focus is given by \(\frac{1}{4a} = \frac{1}{4}\). Since the parabola opens to the right, we'll add this distance to the vertex's x-coordinate. Thus, the focus is at the point \((h + \frac{1}{4}, k) = (2 + \frac{1}{4}, 1) = (\frac{9}{4}, 1)\).

Step 5: Determine the directrix

The directrix is a vertical line that is at a distance of \(\frac{1}{4a}\) to the left of the vertex. So the equation of the directrix is given by \(x = h - \frac{1}{4} = 2 - \frac{1}{4} = \frac{7}{4}\). In conclusion, the parabola's properties are as follows: 1. Vertex: \((2, 1)\) 2. Axis of symmetry: \(x = 2\) 3. Focus: \((\frac{9}{4}, 1)\) 4. Directrix: \(x = \frac{7}{4}\)

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Most popular questions from this chapter

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