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: Equate the second and third parts of the proportion:

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

Simplify the equation:

\[ b^2 - 15 = 2b \]

Rearrange into a quadratic equation:

\[ b^2 - 2b - 15 = 0 \]

Step 2: Solve the quadratic equation for \( b \):

\[ b = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-15)}}{2(1)} = \frac{2 \pm \sqrt{4 + 60}}{2} = \frac{2 \pm \sqrt{64}}{2} = \frac{2 \pm 8}{2} \]

This yields two possible values for \( b \):

\[ b = \frac{2 + 8}{2} = 5 \] or \[ b = \frac{2 - 8}{2} = -3 \]

Step 3: Use the proportion to find \( a \). Equate the first and third parts:

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

Simplify and solve for \( a \):

\[ a + 5 = 2b \Rightarrow a = 2b - 5 \]

Step 4: Calculate the corresponding values of \( a \) and the product \( ab \) for each value of \( b \):

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

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

Correct Option: (C) 33