Consider a pyramid OPQRS located in the first octant `(xge 0, y , z le 0) `with O as origin and OP an OR along the z-axis and the y-axis , respectively , the base OPQR of the pyramid is a squre with OP=3. the point S is directly above the midpoint T of diagonal OQ such that TS=3, then

Options

A. the acute angle between OQ and OS is `(pi)/(3)`
B. the equation of the plane containing the `DeltaOQS` is `x-y=0`
C. the length of the perpendicular from P to the plane containing the `DeltaOQS` is `(3)/(sqrt(2))`
D. the perpendicular distance from O to the straight line containing `RS" is "sqrt((15)/(2))`

Answer

B, C, D

Explanation

Given, square base `OP=OR=3`
`:." "P(3,0,0),R=(0,3,0)`

Also, mid-point of OQ is `T((3)/(2),(3)/(2),0).`
Since, S is directly above the mid-point T of diagonal OQ and ST=3.
i.e.`" "S((3)/(2),(3)/(2),3)`
Here, DR's of OQ(3,3,0) and DR's of `OS((3)/(2),(3)/(2),3).`
`:.costheta=((9)/(2)+(9)/(2))/(sqrt(9+9+0)sqrt((9)/(4)+(9)/(4)+9))=(9)/(sqrt(18).sqrt((27)/(2)))=(1)/(sqrt(3))`
`:.` Option (a) is incorrect.
Now, equation of the plane containing the `DeltaOQS` is
`|{:(x,y,z),(3,3,0),(3//2,3//2,3):}|=0implies|{:(x,y,z),(1,1,0),(1,1,2):}|=0`
`implies" "x(2-0)-y(2-0)+z(1-1)=0`
`implies" "2x-2y=0" or "x-y=0`
`:.` Option (b) is correct.
Now, length of the perpendicular from P(3,0,0) to the plane containing `DeltaOQS` is `(|3-0|)/(sqrt(1+1))=(3)/(sqrt(2))`
`:.` Option (c) is correct.
Here, equation of RS is
`(x-0)/(3//2)=(y-3)/(-3//2)=(z-0)/(3)=lambda`
`implies" "x=(3)/(2)lambda,y=-(3)/(2)lambda+3,z=3lambda`
To find the distance from `O(0,0,0)` to RS.
Let M be the foot of perpendicular.

`((3lambda)/(2),3-(3lambda)/(2),3lambda)((3)/(2),(3)/(2)3)`
`because" "bar(OM)_|_bar(RS)impliesbar(OM).bar(RS)=0`
`implies" "(9lambda)/(4)-(3)/(2)(3-(3lambda)/(2))+3(3lambda)=0implieslambda=(1)/(3)`
`:.M((1)/(2),(5)/(2),1)impliesOM=sqrt((1)/(4)+(25)/(4)+1)=sqrt((30)/(4))=sqrt((15)/(2))`
`:.` Option (d) is correct.