Key Concepts and Formulas

• System of Linear Equations for Planes: A system of three linear equations with three variables represents three planes in $\mathbb{R}^3$. $$\begin{align*} a_1x + b_1y + c_1z &= d_1 \\ a_2x + b_2y + c_2z &= d_2 \\ a_3x + b_3y + c_3z &= d_3 \end{align*}$$
• Conditions for Infinitely Many Solutions (Planes Intersecting in a Line): For the three planes to intersect in a line, the system of linear equations must have infinitely many solutions. This occurs if and only if the determinant of the coefficient matrix ($\Delta$) and all the determinants obtained by replacing a column of coefficients with the constant terms ($\Delta_x, \Delta_y, \Delta_z$) are simultaneously zero.
  • $\Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0$
  • $\Delta_x = \begin{vmatrix} d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3 \end{vmatrix} = 0$
  • $\Delta_y = \begin{vmatrix} a_1 & d_1 & c_1 \\ a_2 & d_2 & c_2 \\ a_3 & d_3 & c_3 \end{vmatrix} = 0$
  • $\Delta_z = \begin{vmatrix} a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\ a_3 & b_3 & d_3 \end{vmatrix} = 0$
• Geometric Interpretation: If $\Delta = 0$ but at least one of $\Delta_x, \Delta_y, \Delta_z$ is non-zero, the system has no solution, implying the planes are parallel or intersect in parallel lines, but not a common line. If all four determinants are zero, the planes intersect in a line or are coincident.

Step-by-Step Solution

Step 1: Set up the System of Equations and Identify Coefficients We are given three planes whose intersection is a line. This means the system of linear equations representing these planes must have infinitely many solutions. The given system is:

$$\begin{align*} x + 4y – 2z &= 1 \quad &(1) \\ x + 7y – 5z &= \beta \quad &(2) \\ x + 5y + \alpha z &= 5 \quad &(3) \end{align*}$$

The coefficient matrix $A$ and the constant vector $D$ are:

$$A = \begin{pmatrix} 1 & 4 & -2 \\ 1 & 7 & -5 \\ 1 & 5 & \alpha \end{pmatrix}, \quad D = \begin{pmatrix} 1 \\ \beta \\ 5 \end{pmatrix}$$

Step 2: Calculate the Determinant of the Coefficient Matrix ($\Delta$) and Find $\alpha$ For the system to have infinitely many solutions, the determinant of the coefficient matrix, $\Delta$, must be zero.

$$\Delta = \begin{vmatrix} 1 & 4 & -2 \\ 1 & 7 & -5 \\ 1 & 5 & \alpha \end{vmatrix}$$

We expand the determinant along the first row:

$$\Delta = 1 \cdot \begin{vmatrix} 7 & -5 \\ 5 & \alpha \end{vmatrix} - 4 \cdot \begin{vmatrix} 1 & -5 \\ 1 & \alpha \end{vmatrix} + (-2) \cdot \begin{vmatrix} 1 & 7 \\ 1 & 5 \end{vmatrix}$$ $$\Delta = 1(7\alpha - (-5)(5)) - 4(1\alpha - (-5)(1)) - 2(1(5) - 7(1))$$ $$\Delta = (7\alpha + 25) - 4(\alpha + 5) - 2(5 - 7)$$ $$\Delta = 7\alpha + 25 - 4\alpha - 20 - 2(-2)$$ $$\Delta = 7\alpha + 25 - 4\alpha - 20 + 4$$ $$\Delta = 3\alpha + 9$$

Setting $\Delta = 0$ for infinitely many solutions:

$$3\alpha + 9 = 0 \implies 3\alpha = -9 \implies \alpha = -3$$

Reasoning: If $\Delta \neq 0$, there would be a unique solution (the planes would intersect at a single point). Since the problem states the intersection is a line, we must have $\Delta = 0$.

Step 3: Calculate $\Delta_z$ and Find $\beta$ Now that we have $\alpha = -3$, the third plane's equation is $x + 5y - 3z = 5$. To ensure infinitely many solutions (and not no solution), we also need $\Delta_x = 0$, $\Delta_y = 0$, and $\Delta_z = 0$. We will use $\Delta_z$ as it contains the unknown parameter $\beta$. $\Delta_z$ is formed by replacing the third column of the coefficient matrix with the constant terms $(1, \beta, 5)^T$:

$$\Delta_z = \begin{vmatrix} 1 & 4 & 1 \\ 1 & 7 & \beta \\ 1 & 5 & 5 \end{vmatrix}$$

Expand $\Delta_z$ along the first row:

$$\Delta_z = 1 \cdot \begin{vmatrix} 7 & \beta \\ 5 & 5 \end{vmatrix} - 4 \cdot \begin{vmatrix} 1 & \beta \\ 1 & 5 \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 7 \\ 1 & 5 \end{vmatrix}$$ $$\Delta_z = 1(7(5) - \beta(5)) - 4(1(5) - \beta(1)) + 1(1(5) - 7(1))$$ $$\Delta_z = (35 - 5\beta) - 4(5 - \beta) + (5 - 7)$$ $$\Delta_z = 35 - 5\beta - 20 + 4\beta - 2$$ $$\Delta_z = 13 - \beta$$

Setting $\Delta_z = 0$ for infinitely many solutions:

$$13 - \beta = 0 \implies \beta = 13$$

Reasoning: If $\Delta = 0$ but any of $\Delta_x, \Delta_y, \Delta_z$ were non-zero, the system would have no solution (the planes would not intersect in a common line). Therefore, all these determinants must be zero for the planes to intersect in a line.

Step 4: Calculate $\alpha + \beta$ Now that we have found the values for $\alpha$ and $\beta$: $\alpha = -3$ $\beta = 13$ We can calculate their sum:

$$\alpha + \beta = -3 + 13 = 10$$

Common Mistakes & Tips

• Confusing No Solution with Infinite Solutions: Remember that $\Delta = 0$ alone is not sufficient for infinitely many solutions. It only implies that there is either no solution or infinitely many solutions. You must check that $\Delta_x, \Delta_y, \Delta_z$ are also zero to confirm infinite solutions.
• Arithmetic Errors in Determinants: Determinant calculations can be tedious and prone to sign errors. Always double-check your calculations, especially when dealing with negative numbers. Using row/column operations to simplify determinants before expansion can sometimes help reduce errors.
• Geometric Interpretation: Visualize what the conditions mean. $\Delta \neq 0$ means a unique point of intersection. $\Delta = 0$ with $\Delta_x, \Delta_y, \Delta_z$ all zero means a line or planes are coincident. $\Delta = 0$ with at least one non-zero $\Delta_i$ means no intersection (parallel planes or parallel lines of intersection).

Summary

To find the values of $\alpha$ and $\beta$ such that the three given planes intersect in a line, we applied the conditions for a system of linear equations to have infinitely many solutions. First, we set the determinant of the coefficient matrix ($\Delta$) to zero, which allowed us to solve for $\alpha = -3$. Next, to ensure infinite solutions (and not no solution), we set one of the other determinants (e.g., $\Delta_z$) to zero, which allowed us to solve for $\beta = 13$. Finally, we calculated the required sum $\alpha + \beta = -3 + 13 = 10$.

The final answer is $\boxed{\text{10}}$, which corresponds to option (A).