Home
Class 12
MATHS
let L be a straight line passing through...

let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes `P_1 : x + 2y-z +1 = 0` and `P_2 : 2x-y + z-1 = 0`, Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane `P_1`. Which of the following points lie(s) on M?

A

`(0,-(5)/(6),-(2)/(3))`

B

`(-(1)/(6),-(1)/(3),(1)/(6))`

C

`(-(5)/(6),0,(1)/(6))`

D

`(-(1)/(3),0,(2)/(3))`

Text Solution

Verified by Experts

The correct Answer is:
A, B
a.b .
Let `vecv` be the vetor along I.
`P_(1):x+2y-Z+1=0 and P_(2):2x-y+z-1=0`
` therefore vecc=|{:(hati,hatj,hatk),( 1,2,-1),( 2,-1,1)}|= hati-3hatj-5hatk`
since I. is through origin any point on line I. is `A(lamda,3lamda ,-5lamda).`Foot of perpendicular from A to `P_(1)` , is
`(h- lamda)/( 1)`=(k+3lamda)/(-1)=-(lamda-6lamda+5lamda+1))/(1+4+1)=-(1)/(6)`
`therefore h= lammda-(1)/(6) ,k =- 3lamda -(1)/(3) ,il=-5 lamda+(1)/(6)`
so, foot ro rpoint on lovus M is `(lamda-(1)/(6),-3lamda-(1)/(3)-(1)/(3),-5lamda+(1)/(6))`
So points (a) and (b) lie on this loucs .
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • JEE 2019

    CENGAGE|Exercise chapter-2|1 Videos

  • JEE 2019

    CENGAGE|Exercise single correct answer type|1 Videos

  • JEE 2019

    CENGAGE|Exercise matching column type|2 Videos

  • INVERSE TRIGONOMETRIC FUNCTIONS

    CENGAGE|Exercise Question Bank|24 Videos

  • LIMITS

    CENGAGE|Exercise Question Bank|30 Videos

Similar Questions

Explore conceptually related problems

let L be a straight line passing through the origin.Suppose that all the points on L are at a constant distance from the two planes P_(1):x+2y-z+1=0 and P_(2):2x-y+z-1=0, Let M be the locus of the feet of the perpendiculars drawn from the points on L.the plane P_(1.) Which of the following points lie(s) on M?

The foot of the perpendicular from the point P(1,3,4) to the plane 2x-y+z+3=0 is

Knowledge Check

  • In R^(2), L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes p _(1) : x+ 2y -z +1=0 and p _(2) : 2x -y + z-1 =0. Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane p _(1). Which of the following points lie (s) on M ?

    A
    `(0,-(5)/(6) - (2)/(3))`
    B
    `(-(1)/(6),- (1)/(3), (1)/(6))`
    C
    `(- (5)/(6), 0, (1)/(6))`
    D
    `(-(1)/(3), 0,(2)/(3))`

  • Let two planes p _(1): 2x -y + z =2 and p _(2) : x + 2y - z=3 are given : Equation of the plane which passes through the point (-1,3,2) and is perpendicular to each of the plane is:

    A
    `x + 3y - 5z +2 =0`
    B
    ` x + 3y + 5z + 18 =0`
    C
    `x - 3y - 5z + 20=0`
    D
    `x - 3y + 5z =0`

  • Let L: x-2y+4z=9 be a plane, P(2,1,-1) be a point and O, be the Origin. Q is the foot of the perpendicular from P on the plane L, then (OP)^(2)+(PQ)^(2) is equal to

    A
    `(190)/(21)`
    B
    `(43)/(21)`
    C
    `(295)/(21)`
    D
    `(211)/(21)`

    • Similar Questions

    Explore conceptually related problems

    Let the acute angle bisector of the two planes x - 2y - 2z + 1 = 0 and 2x - 3y - 6 + 1= 0 be the plane P. Then which of the following points lies on P ?

    Let the acute angle bisector of the two planes x - 2y - 2z + 1 = 0 and 2x - 3y - 6 z+ 1= 0 be the plane P. Then which of the following points lies on P ?

    Find the coordinates of the foot of the perpendicular and the perpendicular distance from the point P(3,2,1) to the plane 2x-y+z+1=0 .

    Find the coordinates of the foot of the perpendicular and the perpendicular distance from the point P(3,2,1) to the plane 2x-y+z+1=0 Find also the image of the point P in the plane.

    Let L_(1) be a straight line passing through the origin and L_(2) be the straight line x+y=1 if the intercepts made by the circle x^(2)+y^(2)-x+3y=0 on L_(1) and L_(2) are equal,then which of the following equations can represent L_(1)?

    Home
    Profile