Single correct option
Real numbers x and y satisfy 3x + 4y = 25. What is the minimum possible value of x² + y²?
- A. 5
- B. 25
- C. 625/7
- D. 12.5
Show answer and solution
Answer: (B) 25
x² + y² is the square of the distance of (x, y) from the origin, and (x, y) lies on the line 3x + 4y = 25.
The shortest distance from the origin to this line = |25|/√(3² + 4²) = 25/5 = 5.
Minimum of x² + y² = 5² = 25, attained at (3, 4).
Trap: 5 is the minimum distance, not its square. (Cauchy–Schwarz gives the same: (3x + 4y)² ≤ 25(x² + y²).)
The shortest distance from the origin to this line = |25|/√(3² + 4²) = 25/5 = 5.
Minimum of x² + y² = 5² = 25, attained at (3, 4).
Trap: 5 is the minimum distance, not its square. (Cauchy–Schwarz gives the same: (3x + 4y)² ≤ 25(x² + y²).)
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