find the ratio in which the join of A (2,1,5) b(3,4,3) is divided by the plane 2x+2y-2z =1..also find coordinates of the pt. of division ?
Suppose the plane 2x + 2y – 2z = 1 meets the line joining the points A(2, 1, 5) and B(3, 4, 3) at the point C, and C divides AB in the ratio K: 1.
Then the coordinates of C are (3K +2/K + 1,4K +1/ K + 1,3K + 5/ K + 1)
But the point C lies on the plane 2x + 2y – 2z = 1. So its coordinates must satisfy the equation of this plane.
∴ 2(3K +2/K + 1) + 2(4K +1/ K + 1) + 2(3K + 5/ K + 1)
or 6λ + 4 + 8λ + 2 - 6λ - 10 = λ + 1 or 7λ = 5 ⇒ λ = 5/7
∴ required ratio is 5 : 7. Putting λ = 5:7 in (1)
the coordinates of the point of division C are (29/12,9/4,25/6)
Then the coordinates of C are (3K +2/K + 1,4K +1/ K + 1,3K + 5/ K + 1)
But the point C lies on the plane 2x + 2y – 2z = 1. So its coordinates must satisfy the equation of this plane.
∴ 2(3K +2/K + 1) + 2(4K +1/ K + 1) + 2(3K + 5/ K + 1)
or 6λ + 4 + 8λ + 2 - 6λ - 10 = λ + 1 or 7λ = 5 ⇒ λ = 5/7
∴ required ratio is 5 : 7. Putting λ = 5:7 in (1)
the coordinates of the point of division C are (29/12,9/4,25/6)
