Home
Class 12
MATHS
In R^(3), consider the planes P(1):y=0 a...

In `R^(3)`, consider the planes `P_(1):y=0` and `P_(2),x+z=1.` Let `P_(3)` be a plane, different from `P_(1)` and `P_(2)` which passes through the intersection of `P_(1)` and `P_(2)`, If the distance of the point (0,1,0) from `P_(3)` is 1 and the distance of a point `(alpha,beta,gamma)` from `P_(3)` is 2, then which of the following relation(s) is/are true? (a) `2alpha + beta + 2gamma +2 = 0 ` (b) `2alpha -beta + 2gamma +4=0` (c) `2alpha + beta - 2gamma- 10 =0` (d) `2alpha- beta+ 2gamma-8=0`

A

`2alpha + beta + 2gamma +2 = 0 `

B

`2alpha -beta + 2gamma +4=0`

C

`2alpha + beta - 2gamma- 10 =0`

D

`2alpha- beta+ 2gamma-8=0`

Text Solution

Verified by Experts

The correct Answer is:
b, d
Clearly, plane `P_(3)` is `P_(2)+ lamdaP_(1)=0`.
`rArr" "x+lamday+z-1=0`
Distance of this plane from point `(0, 1, 0)` is 1.
`rArr " "(0+lamda+0-1)/(sqrt(1+lamda^(2)+1))= pm 1`
`therefore " "lamda= - (1)/(2)`
Thus, equation of `P_(3)` is `2x-y+2z-2=0`.
Distance of this plane from point `(alpha, beta, gamma)` is 2.
`rArr" "|(2alpha-beta+2gamma-2)/(3)|=2`
`rArr" "2alpha-beta+2gamma= 2pm 6`
Thus, option (b), (d) are correct.
Promotional Banner

Topper's Solved these Questions

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE PUBLICATION|Exercise ARCHIVES REASONING TYPE|2 Videos

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE PUBLICATION|Exercise ARCHIVES LINKED COMPREHENSION TYPE|3 Videos

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE PUBLICATION|Exercise ARCHIVES SINGLE CORRECT ANSWER TYPE|9 Videos

  • THREE DIMENSIONAL GEOMETRY

    CENGAGE PUBLICATION|Exercise All Questions|291 Videos

  • TRIGONOMETRIC EQUATIONS

    CENGAGE PUBLICATION|Exercise Archives (Numerical value type)|4 Videos

Similar Questions

Explore conceptually related problems

In R, consider the planes P_(1): y = 0 and P_(2) : x +z = 1 . Let P_(3) be a plane, different from P_(1) and P_(2) , which passes through the intersection of P_(1) and P_(2) . If the distance of the point (0, 1,1) from P_(3) is 1 and the distance of a point (alpha, beta, gamma) " from " P_(3) is 2, then which of the following relations is (are) true?

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

The equation of a plane which passes through the point of intersection of lines (x-1)/(3)=(y-2)/(1)=(z-3)/(2) , and (x-3)/(1)=(y-1)/(2)=(z-2)/(3) and at greatest distance from point (0,0,0) is

The point P is equidistant from the points (-1,4,2), (2,1,2) ,(3,2,2) and (0,5,6) , find the coordinates of the point P.

The points (p+1,1), (2p+1,3) and (2p+2, 2p) are collinear, if

Find the equation of the plane passing through the point A(1,2,1) and perpendicular to the line joining the points P(1,4,2) and Q (2,3,5) . Also , find the distance of this plane from the line (x+3)/(2)=(y-5)/(-1)=(z-7)/(-1) .

A variable line passes through a fixed point P. The algebraic sum of the perpendiculars drawn from the points (2,0), (0,2) and (1,1) on the line is zero. Find the coordinate of the point P.

A line through point A(1,3) and parallel to the line x-y+1 = 0 meets the line 2x-3y + 9 = 0 at point P. Find distance AP without finding point P.

A point P(x,y) moves in the xy - plane in such a way that its distance from the point (0,4) is equal to (2)/(3) rd of its distance from the x axis , find the equation to the locus of P.

Vertices of a parallelogram taken in order are A( 2,-1,4)B(1,0,-1)C( 1,2,3) and D. Distance of the point P ( 8, 2,-12) from the plane of the parallelogram is