Given planes are 4x + 4y – 5z – 12 = 0 ..... (1)
and 8x + 12y – 13z – 32 = 0 ..... (2)
Let l, m, n be the direction ratios of the line of intersection :
then 4l + 4m – 5n = 0 ..... (3)
and 8l + 12m – 13n = 0
Hence direction ratios of line of intersection are 2, 3, 4.
Here 4 ≠ 0, therefore line of intersection is not parallel to xy-plane.
Let the line of intersection meet the xy-plane at P (α, β, 0).
Then P lies on planes (1) and (2)
∴ 4α + 4β + 12 = 0
or, α + β – 3 = 0 ..... (5)
and 8α + 12β – 32 = 0
or, 2α + 3β – 8 = 0 ..... (6)
Solving (5) and (6), we get
Hence equation of line of intersection in symmetrical form is
