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

Concept: Standard form: \[ (x-h)^2 = 4p(y-k) \]

Step 1: Complete square.
\[ x^2 - 4x + 8y + 4 = 0 \] \[ (x-2)^2 - 4 + 8y + 4 = 0 \] \[ (x-2)^2 + 8y = 0 \] \[ (x-2)^2 = -8y \]

Step 2: Compare with standard form.
\[ (x-2)^2 = -8(y-0) \Rightarrow 4p = -8 \Rightarrow p = -2 \]

Step 3: Find focus.
\[ (h, k+p) = (2, 0 - 2) = (2,-2) \]