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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.

Expert verified
The properties of the given parabola are: Vertex: $$(2, 1)$$, Axis of symmetry: $$x = 2$$, Focus: $$(\frac{9}{4}, 1)$$, and Directrix: $$x = \frac{7}{4}$$.
See the step by step solution

## 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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