Question

For some real numbers \(a\) and \(b\) , the system of equations \(x + y = 4\) and \((a+5)x+(b^2-15)y = 8b\) has infinitely many solutions for \(x\) and \(y\). Then, the maximum possible value of \(ab\) is

• A. 15
• B. 55
• C. 33
• D. 25

Answer 1

The Correct Option is C

Given:
The system of equations has infinitely many solutions:

• \( x + y = 4 \)                 (1) 
• \( (a + 5)x + (b^2 - 15)y = 8b \)   (2)

Condition for infinitely many solutions:
Two linear equations have infinitely many solutions if their corresponding coefficients are proportional:

\[ \frac{a + 5}{1} = \frac{b^2 - 15}{1} = \frac{8b}{4} \]

Step 1: Equating the second and third terms:

\[ \frac{b^2 - 15}{1} = \frac{8b}{4} \Rightarrow b^2 - 15 = 2b \Rightarrow b^2 - 2b - 15 = 0 \]

Step 2: Solving the quadratic equation:

\[ b = \frac{2 \pm \sqrt{(-2)^2 + 4 \cdot 1 \cdot 15}}{2} = \frac{2 \pm \sqrt{4 + 60}}{2} = \frac{2 \pm \sqrt{64}}{2} = \frac{2 \pm 8}{2} \] So, \( b = 5 \) or \( b = -3 \)

Step 3: Now use the proportion to find a:

\[ \frac{a + 5}{1} = \frac{8b}{4} = 2b \Rightarrow a + 5 = 2b \Rightarrow a = 2b - 5 \]

Step 4: Find values of a corresponding to each b:

• When \( b = 5 \):
  \[ a = 2 \cdot 5 - 5 = 10 - 5 = 5 \Rightarrow ab = 5 \cdot 5 = 25 \]
• When \( b = -3 \):
  \[ a = 2 \cdot (-3) - 5 = -6 - 5 = -11 \Rightarrow ab = (-11) \cdot (-3) = 33 \]

Final Answer:
The maximum value of \( ab \) is 33.

Correct Option: (C) 33