If the straight line (x - 1)/k = (y - 2)/2 = (z - 3)/3 and (x - 2)/3 =

1.	If the straight line (x - 1)/k = (y - 2)/2 = (z - 3)/3 and (x - 2)/3 = (y - 3)/k = (z - 1)/2 intersect at a point, then find the value of integer k.
Answer» Given lines are (x - 1)/k = (y - 2)/2 = (z - 3)/3 = λ ...(i) and (x - 2)/3 = (y - 3)/k = (z - 1)/2 = μ ...(ii) Any point on line (i) is (kλ + 1,2λ + 2,3λ + 3) and any point on line (ii) (3μ + 2,μk + 3,2μ + 1) If these line interest, these points must coincide for some value of and . So, λk + 1 = 3μ + 2, 2λ + 2 = μk + 3, 3λ + 3 = 2μ + 1 λk - 3λ - 1 = 0, 2λ - μk - 1 = 0, 3λ - 2μ + 2 = 0 taking first two, we get λ/(3 - k) = μ/(-2 + k) = 1/(-k² + 6) λ = (3 - k)/(6 - k²), μ = (k - 2)/(6 - k²) Putting these value of λ & μ in 3λ - 2μ + 2 = 0 3(3 - k)/(6 - k²) - 2(k - 2)/(6 - k²) + 2 = 0 9 - 3k - 2k + 4 + 12 - 2k² = 0 2k² + 5k - 25 = 0 2k(k + 5) - 5(k + 5) = 0 (2k - 5)(k - 5) = 0 k = 5/2, -5 k = -5 satisfy both the equation. Hence, k = - 5 is require value.

If the straight line (x - 1)/k = (y - 2)/2 = (z - 3)/3 and (x - 2)/3 = (y - 3)/k = (z - 1)/2 intersect at a point, then find the value of integer k.

Answer»

Given lines are (x - 1)/k = (y - 2)/2 = (z - 3)/3 = λ ...(i)

and (x - 2)/3 = (y - 3)/k = (z - 1)/2 = μ ...(ii)

Any point on line (i) is (kλ + 1,2λ + 2,3λ + 3) and any point on line (ii) (3μ + 2,μk + 3,2μ + 1)

If these line interest, these points must coincide for some value of and .

So, λk + 1 = 3μ + 2, 2λ + 2 = μk + 3, 3λ + 3 = 2μ + 1

λk - 3λ - 1 = 0, 2λ - μk - 1 = 0, 3λ - 2μ + 2 = 0

taking first two, we get

λ/(3 - k) = μ/(-2 + k) = 1/(-k² + 6)

λ = (3 - k)/(6 - k²), μ = (k - 2)/(6 - k²)

Putting these value of λ & μ in 3λ - 2μ + 2 = 0

3(3 - k)/(6 - k²) - 2(k - 2)/(6 - k²) + 2 = 0

9 - 3k - 2k + 4 + 12 - 2k² = 0

2k² + 5k - 25 = 0

2k(k + 5) - 5(k + 5) = 0

(2k - 5)(k - 5) = 0

k = 5/2, -5

k = -5 satisfy both the equation.

Hence, k = - 5 is require value.