Perpendiculars are drawn from points on the line x + 2 2 = y + 1 - 1 = z 3 to the plane x + y + z = 3 . The…

Perpendiculars are drawn from points on the line  x + 2 2 = y + 1 - 1 = z 3  to the plane x+y+z=3. The feet of perpendiculars lie on the line
  1. x 5 = y - 1 8 = z - 2 - 1 3
  2. x 2 = y - 1 3 = z - 2 - 5
  3. x 4 = y - 1 3 = z - 2 - 7
  4. x 2 = y - 1 - 7 = z - 2 5

Solution

The equation od given line is  x + 2 2 = y + 1 - 1 = z 3 = λ
Now any two points on this line are
A-2,-1,0,B0,-2,3  forλ=0, 1
Let foot of perpendicular from A(-2,-1,0) on plane is α ,  β ,  γ
⇒α+21=β+11=γ-01=μsay
Also the point lies on the plane, hence  α + β + γ = 3
⇒μ-2+μ-1+μ=3⇒μ=2

Hence the coordinates of M are
⇒M0, 1, 2
Similarly foot of perpendicular from B(0, -2, 3) on plane is N 2 3 , - 4 3 , 1 1 3
So the required equation of MN is x-023=y-1-73=z-253

⇒x-02=y-1-7=z-25

Asked in: JEE Advanced 2013 (Paper 1)

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