# Show that the lines (x+1)/3=(y+3)/5=(z+5)/7 and (x−2)/1=(y−4)/3=(z−6)/5 intersect. Also find their point of intersection - Mathematics

Show that the lines (x+1)/3=(y+3)/5=(z+5)/7 and (x−2)/1=(y−4)/3=(z−6)/5 intersect. Also find their point of intersection

#### Solution

We have:

(x+1)/3=(y+3)/5=(z+5)/7=λ (say)

x=3λ1, y=5λ3 and z=7λ5
So, the coordinates of a general point on this line are (3λ1, 5λ3, 7λ5).
The equation of the second line is given below:

(x−2)/1=(y−4)/3=(z−6)/5=μ (say)

x=μ+2, y=3μ+4 and z=5μ+6

So, the coordinates of a general point on this line are (μ+2, 3μ+4, 5μ+6).
If the lines intersect, then they have a common point.
So, for some values of λ and μ, we must have:

3λ1=μ+2, 5λ3=3μ+4 and 7λ5=5μ+6

3λμ=3, 5λ3μ=7 and 7λ5μ=11

Solving the first two equations,  3λμ=3 and 5λ3μ=7, we get:

λ=1/2 and μ=3/2

Since λ=1/2 and μ=3/2 satisfy the third equation, 7λ5μ=11, the given lines intersect each other.

When λ=1/2 in (3λ1, 5λ3, 7λ5), the coordinates of the required point of intersection are (1/2, 1/2, 3/2)

Is there an error in this question or solution?
2013-2014 (March) Delhi Set 1

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