find k so that the lines: (1-2x)/3 = (y-2)/2k = (z-3)/2 and (3x-1)/2k = (y-5)/1 = (5-2z)/3 are at right - Maths - Three Dimensional Geometry

Question

find k so that the lines: (1-2x)/3 = (y-2)/2k = (z-3)/2 and
(3x-1)/2k = (y-5)/1 = (5-2z)/3 are at right angles

Manbar Singh · Accepted Answer

The given equation of the lines are :1 - 2x3 = y-22k = z - 32⇒x - 12-32 = y - 22k = z-32    
1The dr's of the line 1 are :a1, b1, c1 = -32, 2k, 2and 3x - 12k = y-51 = 5-2z3⇒x - 132k3 = y-51 = z - 5-32    
2The dr's of the line 2 are :a2, b2, c2 = 2k3, 1, -32Since, line 1 and line 2 are ⊥ to each other, then     a1a2 + b1b2 + c1c2 =0⇒-32×2k3 + 2k×1 + 2×-32 = 0⇒-k + 2k - 3 = 0⇒k = 3

find k so that the lines: (1-2x)/3 = (y-2)/2k = (z-3)/2 and (3x-1)/2k = (y-5)/1 = (5-2z)/3 are at right angles