  Question

# ABCD is a regular tetrahedron. M(0,0,0) is the midpoint of the altitude DN of the tetrahedron. Coordinates of the points A and B are (0,0,−2) and (b,b,0) respectively (with b>0). Also x - coordinate of the point C is given to be positive. Then which of the following is/are correct?

A
Image of point D in the plane generated by ABC is (22,0,2)
B
Image of point D in the plane generated by ABC is (2,0,22)
C
Volume of tetrahedron ACBM is 423
D
Volume of tetrahedron ACBM is 43

Solution

## The correct options are A Image of point D in the plane generated by ABC is (2√2,0,−2) D Volume of tetrahedron ACBM is 43AB=BC=CD=AD=a a=2√2 MA=MB=MC=2    (Using symmetry) MB=2⇒2b2=4⇒b=√2 B(√2,√2,0) MA2+MB2=AB2⇒−−→MA⊥−−→MB Similarly −−→MA⊥−−→MC,−−→MB⊥−−→MC ⇒−−→MA.−−→MC⇒−2z=0−−→MB.−−→MC⇒√2x+√2y=0⇒y=−x & z=0 ∴C(x,−x,0) Also MC2=2x2=MA2=4⇒x=√2  (x>0) So C(√2,−√2,0) Coordinates of N are (2√23,0,−23) and hence D is (−2√23,0,23).DN is normal to the plane ABC. N is mid point of DD′ so D′=(2√2,0,−2) Since MA,MB,MC are orthogonal triads Volume of tetrahedron is= 16|MA|.|MB|.|MC|=43.  Suggest corrections   