Question

If \(x - 2y = 4\), then the minimum value of \(xy\) is

• A. \(-2\)
• B. 0
• C. 1
• D. \(-3\)

Hint:

For a quadratic expression \(ay^2 + by + c\) with \(a > 0\), minimum occurs at vertex \(y = -\frac{b}{2a}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Express \(x\) in terms of \(y\) and form a quadratic.

Step 2: Detailed Explanation:
\(x = 2y + 4\).
Let \(P = xy = y(2y + 4) = 2y^2 + 4y = 2(y^2 + 2y) = 2[(y+1)^2 - 1] = 2(y+1)^2 - 2\).
Minimum occurs when \((y+1)^2 = 0\), i.e., \(y = -1\), \(x = 2\).
Minimum value = \(-2\).

Step 3: Final Answer:
Option (A) \(-2\).