The line passing through the points (a, 1,6) and (3,4, b) crosses the yz -plane at (0, 17 2 , -13 2 ) , then the value of (3 a+4 b) is

Step-by-step solution

Given: The line through P₁ = (a, 1, 6) and P₂ = (3, 4, b) intersects the yz-plane at P₃ = (0, 17 2 , - 13 2 ) . We determine a and b using the section formula. Since P₃ lies on the yz-plane, its x-coordinate is zero. From the section formula: 0 = 3 + a + 1 3 + a = 0 Using the y-coordinate: 17 2 = 4 + 1 + 1 17 + 17 = 8 + 2 9 = -15 = - 5 3 Substituting into the x-coordinate relation: 3(- 5 3 ) + a = 0 a = 5 Using the z-coordinate: - 13 2 = - 5 3 b + 6 - 5 3 + 1 = - 5b 3 + 6 - 2 3 = -5b + 18 -2 Multiplying both sides