If the two lines l₁:(x - 2)/3 = (y + 1)/-2, z = 2 and l₂:(x - 1)/1 = (2y + 3)/α = (z + 5)/2 are perpendicular, then an angle between the lines l₂ and l₃:(1 - x)/3 = (2y - 1)/(- 4) = z/4 is :

Solution

Maths sir · 4 steps

Before reading off direction ratios, make the coefficients of x, y and z all equal to +1.

1. Standard form first. Direction ratios come from the denominators only when the numerators are x - a, y - b, z - c. So write 2y + 3 as 2(y + 3/2) and 1 - x as -(x - 1) before you read anything off.
2. Find alpha. l₁ has direction (3, -2, 0), since z = 2 is fixed. l₂ becomes (x-1)/1 = (y + 3/2)/(α/2) = (z+5)/2, with direction (1, α/2, 2). Perpendicular lines give 3 - α + 0 = 0, so α = 3.
3. Directions of l2, l3. With α = 3, l₂ has direction (1, 3/2, 2), or (2, 3, 4). l₃ becomes (x-1)/-3 = (y - 1/2)/-2 = z/4, with direction (-3, -2, 4). Each vector has magnitude √29.

   cos θ = ((2)(-3) + (3)(-2) + (4)(4))/(√29 √29) = 4/29

4. So. θ = cos⁻¹(4/29), which is the same angle as sec⁻¹(29/4). The answer is (b).

The idea to keep. Always rewrite a line as (x-a)/l = (y-b)/m = (z-c)/n before reading direction ratios. Watch for 2y or 1 - x in the numerators, and remember that cos⁻¹ p/q = sec⁻¹ q/p.

Answer from the NTA final answer key. Solution by PrepWiser.