Question

Find the focus of the parabola \( x^2 - 4x + 8y + 4 = 0 \).

• A. \( (2, 1) \)
• B. \( (1, 2) \)
• C. \( (-2, 1) \)
• D. \( (2, -1) \)

Hint:

When completing the square to put a parabola in standard form, remember that the focus is at a distance \( p \) from the vertex along the axis of symmetry.

Answer 1

The Correct Option is A

We are given the equation of a parabola: \[ x^2 - 4x + 8y + 4 = 0 \] We need to find the focus of this parabola.

1. Step 1: Rearrange the equation into standard form First, group the \( x \)-terms together: \[ x^2 - 4x = -8y - 4 \] Now, complete the square on the left-hand side: \[ x^2 - 4x + 4 = -8y - 4 + 4 \] This simplifies to: \[ (x - 2)^2 = -8(y + \frac{1}{2}) \]

2. Step 2: Identify the standard form of a parabola The equation now resembles the standard form of a parabola: \[ (x - h)^2 = 4p(y - k) \] where \( (h, k) \) is the vertex of the parabola and \( p \) is the distance from the vertex to the focus.

3. Step 3: Identify the vertex and focus Comparing the equation \( (x - 2)^2 = -8(y + \frac{1}{2}) \) with the standard form, we can see that: \[ h = 2, \quad k = -\frac{1}{2}, \quad 4p = -8 \quad \Rightarrow \quad p = -2 \] The vertex is at \( (2, -\frac{1}{2}) \), and the focus is \( 2 \) units below the vertex because \( p = -2 \). Therefore, the focus is at \( (2, 1) \). Thus, the focus of the parabola is \( (2, 1) \).