Question
Show that the lines:
are co-planar, find their common point and the equation of the plane in which way they lie.
Solution
The equations of lines are
...(1)
...(2)
Any point on the line (1) is (4 r + 5, 4 r + 7, – 5 r – 3)
It lies on the line (2), if
i.e., if
...(3)
Taking
we get
Substituting this value of r in (3), we get,
...(1) ...(2)Any point on the line (1) is (4 r + 5, 4 r + 7, – 5 r – 3)
It lies on the line (2), if
i.e., if
...(3)Taking
we getSubstituting this value of r in (3), we get,
or – 1 = – 1 = – 1, which is true
∴ the lines intersect, and ∴ are coplanar
The point of intersection of lines is (–4 + 5, –4 + 7, 5 – 3) i.e. (1,3,2)
The equation of plane in which given lines lie is
or 
or (x – 5) (12+ 5)–(y–7) (12 + 25) + (z + 3) (4 – 28) = 0
or 17 (x – 5) – 47 (y – 7) – 24 (z + 3) = 0
or 17x – 47 y – 24 z + 172 = 0
which is required equation of plane.
