Question

The coordinates of the focus of the parabola given by the equation \(4y - x^2 + 4x - 12 = 0\) are

• A. (2,3)
• B. (3,2)
• C. (3,3)
• D. (3,7)
• E. (7,2)

Hint:

Focus and Vertex share the same x-coordinate in parabolas opening upwards or downwards. Since the x-term is squared, find its completed square form to get the x-coordinate of the focus immediately.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We must convert the given general equation into the standard vertex form \((x-h)^2 = 4a(y-k)\) to identify the vertex \((h,k)\) and the focal distance \(a\).

Step 2: Key Formula or Approach:
For parabola \((x-h)^2 = 4a(y-k)\), the focus is \((h, k+a)\).

Step 3: Detailed Explanation:
The equation is \(x^2 - 4x = 4y - 12\).
Complete the square for the x-terms:
\[ (x^2 - 4x + 4) - 4 = 4y - 12 \]
\[ (x - 2)^2 = 4y - 8 \]
\[ (x - 2)^2 = 4(y - 2) \]
Comparing with \((x-h)^2 = 4a(y-k)\):
\(h = 2, k = 2\)
\(4a = 4 \implies a = 1\)
The focus is \((h, k+a) = (2, 2+1) = (2, 3)\).

Step 4: Final Answer:
The coordinates of the focus are (2, 3).