Question

Let the system of equations be: $ 2x + 3y + 5z = 9, $ $ 7x + 3y - 2z = 8, $ $ 12x + 3y - (4 + \lambda)z = 16 - \mu, $ which has infinitely many solutions. Then the radius of the circle centered at $ (\lambda, \mu) $ and touching the line $ 4x = 3y $ is:

• A. \( \frac{17}{5} \)
• B. \( \frac{7}{5} \)
• C. 7
• D. \( \frac{21}{5} \)

Hint:

When solving for the radius of a circle, use the formula for the perpendicular distance from a point to a line. Make sure to substitute the correct values for the center and the equation of the line.

Answer 1

The Correct Option is B

Step 1: Condition for infinite solutions.
A system of equations has infinitely many solutions if the determinant of its coefficient matrix is zero. The coefficient matrix for the given system is: \[ \begin{pmatrix} 2 & 3 & 5 7 & 3 & -2 12 & 3 & -(4 + \lambda) \end{pmatrix}. \] The determinant is calculated as: \[ \text{det} = 2 \left( \begin{vmatrix} 3 & -2 3 & -(4 + \lambda) \end{vmatrix} \right) - 3 \left( \begin{vmatrix} 7 & -2 \\ 12 & -(4 + \lambda) \end{vmatrix} \right) + 5 \left( \begin{vmatrix} 7 & 3 \\ 12 & 3 \end{vmatrix} \right). \] Setting the determinant to zero yields values for \( \lambda \) and \( \mu \).
Step 2: Solve for \( \lambda \) and \( \mu \).
The determinant evaluation proceeds as follows: \[ \text{det} = 12(21) - 3(39) - (\lambda + 4)(-15) = 0, \] \[ \Rightarrow -252 + 117 + 15(1 + 4) = 0, \] \[ \Rightarrow 15\lambda + 177 - 252 = 0, \] \[ \Rightarrow 15\lambda - 75 = 0 \Rightarrow \lambda = 5. \] For \( \mu \), the matrix is: \[ \begin{pmatrix} 9 & 3 & 5 8 & 3 & -2 16 & 3 & -(4 + \mu) \end{pmatrix}. \] Solving for \( \mu \) yields \( \mu = 9 \). 
Step 3: Determine the circle's radius.
The circle's center is at \( (\lambda, \mu) = (5, 9) \). The radius is the perpendicular distance from this center to the line \( 4x = 3y \). 
The distance formula from a point \( (x_1, y_1) \) to a line \( ax + by + c = 0 \) is: \[ \text{Distance} = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}. \] With \( a = 4, b = -3, c = 0, x_1 = 5, y_1 = 9 \), the distance is: \[ \text{Distance} = \frac{|4(5) - 3(9) + 0|}{\sqrt{4^2 + (-3)^2}} = \frac{|20 - 27|}{5} = \frac{7}{5}. \] 
Therefore, the radius of the circle is \( \frac{7}{5} \).