Q. 215.0( 2 Votes )

Find the equation

Answer :


Let first line be and second


Let A be the point (1, 1, 1) and B and C be points on first line and second line respectively


Consider first line:


x + 2 = λ and y – 3 = 2λ and z + 1 = 4λ


x = λ – 2 and y = 2λ + 3 and z = 4λ – 1


Hence the point B on line will have coordinates (λ – 2, 2λ + 3, 4λ – 1)


Let us write the equation of line AB we know a point on the line (1, 1, 1) we need the direction which we will get by writing the vector




Hence the equation of line AB will be



Consider second line


x – 1 = 2μ and y – 2 = 3μ and z – 3 = 4μ


x = 2μ + 1 and y = 3μ + 2 and z = 4μ + 3


Hence the point C on line will have coordinates (2μ + 1, 3μ + 2, 4μ + 3)


Let us write the equation of line AC we know a point on the line (1, 1, 1) we need the direction which we will get by writing the vector




Hence the equation of line AC will be



Now as we have to write equation of line which intersects first and second line and also passes through A


Which means we have to write equation of line which passes through A, B and C


AB intersects first line and AC intersects second line


AB and AC will represent the same line if the direction ratios of equation (i) and (ii) are in proportion



λ – 3 = 2μk …(p)


2λ + 2 = k (3μ + 1)


2λ + 2 = 3μk + k …(q)


4λ – 2 = k (4μ + 2)


2λ – 1 = k (2μ + 1)


2λ – 1 = 2μk + k …(r)


Rewrite equation (q) as


2λ + 2 = (2μk + k) + μk


Using (p) and (r)


2λ + 2 = 2λ – 1 +



6 = λ – 3


λ = 9


Put λ = 9 in (p)


9 – 3 = 2μk


2μk = 6


Put λ = 9 and 2μk = 6 in (r)


2(9) – 1 = 6 + k


k = 11


Put k in 2μk = 6


2μ (11) = 6



Put value of λ and μ in (i) and (ii) respectively which are representing the equation of same line


In (i)




In (ii)





Hence equation of line is


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