Question

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

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

Hint:

Always complete the square first to convert into standard parabola form.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To find the focus of a parabola, we must first convert its equation into the standard vertex form. This process involves completing the square.
Step 2: Key Formula or Approach:
The standard equation for a vertical parabola is \((x-h)^2 = 4p(y-k)\), where \((h, k)\) is the vertex. The focus for such a parabola is located at \((h, k+p)\). We will rearrange the given equation to match this standard form.
Step 3: Detailed Explanation:
The given equation is: \[ x^2 - 4x + 8y + 4 = 0 \] 1. Isolate the x-terms and complete the square. Move the y-term and the constant to the other side: \[ x^2 - 4x = -8y - 4 \] To complete the square for \(x^2 - 4x\), we take half of the coefficient of x (-4), which is -2, and square it to get 4. Add this value to both sides: \[ (x^2 - 4x + 4) = -8y - 4 + 4 \] Factor the left side, which is now a perfect square: \[ (x - 2)^2 = -8y \] 2. Identify the vertex (h, k) and the parameter p. We write the equation in the standard form \((x-h)^2 = 4p(y-k)\): \[ (x - 2)^2 = -8(y - 0) \] By comparing the forms, we can identify:
\(h = 2\)
\(k = 0\)
\(4p = -8 \implies p = -2\)
The vertex of the parabola is \((h, k) = (2, 0)\). Since p is negative, the parabola opens downwards. 3. Find the focus. The focus of a vertical parabola is at the point \((h, k+p)\). \[ \text{Focus} = (2, 0 + (-2)) = (2, -2) \] Step 4: Final Answer:
The focus of the parabola is (2, -2).