<p>Three planes are given by:</p><p><math>x + 4y - 2z = 1</math></p><p><math>x + 7y - 5z = b</math></p><p><math>x + 5y + az = 5</math></p><p>If their intersection is a line in <math>\mathbb{R}^3</math>, find the values of <math>a</math> and <math>b</math>.</p>
Step-by-Step Solution
Key Concept: For three planes to intersect in a line, the coefficient matrix determinant must be zero and at least one other determinant formed with constants must also be zero.
<p><strong>For intersection to be a line:</strong> <math>D = 0</math> and <math>D_1 = 0</math>, <math>D_2 = 0</math></p><p><strong>Step 1:</strong> <math>D = \begin{vmatrix} 1 & 4 & -2 \\ 1 & 7 & -5 \\ 1 & 5 & a \end{vmatrix} = 0</math></p><p><math>1(7a + 25) - 4(a + 5) - 2(5 - 7) = 0</math></p><p><math>7a + 25 - 4a - 20 + 4 = 0</math></p><p><math>3a + 9 = 0 \Rightarrow a = -3</math></p><p><strong>Step 2:</strong> <math>D_2 = \begin{vmatrix} 1 & 4 & 1 \\ 1 & 7 & b \\ 1 & 5 & 5 \end{vmatrix} = 0</math></p><p><math>1(35 - 5b) - 4(5 - b) + 1(5 - 7) = 0</math></p><p><math>35 - 5b - 20 + 4b - 2 = 0</math></p><p><math>b = 13</math></p><p><strong>Therefore:</strong> <math>a + b = -3 + 13 = 10</math></p>
Correct Answer: a = -3, b = 13