`A B C D` is a tetrahedron and `O` is any point. If the lines joining `O` to the vrticfes meet the opposite faces at `P ,Q ,Ra n dS ,` prove that `(O P)/(A P)+(O Q)/(B Q)+(O R)/(C R)+(O S)/(D S)=1.`

In Fig. 4.90, if P S Q R , prove that P O S ~ R O Q . (FIGURE)

A B C D is a parallelogram whose A C\ a n d\ B D intersect at Odot A line through O intersects A B\ a t\ P\ a n d\ D C\ at Qdot Prove that a r\ (\ P O A)=\ a r\ (\ Q O C)dot

A B C D is a parallelogram whose diagonals A C and B D intersect at Odot A Line through O intersects A B at P and D C at Qdot Prove that a r( P O A)=a r( Q O C)dot

O is any point on the diagonal B D of the parallelogram A B C Ddot Prove that a r( O A B)=a r( O B C)

In Fig. 4.90, if P O S ~ R O Q , prove that P S Q R . (FIGURE)

Find the value of lambda so that the points P ,Q ,R and S on the sides O A ,O B ,OC and A B , respectively, of a regular tetrahedron O A B C are coplanar. It is given that (O P)/(O A)=1/3,(O Q)/(O B)=1/2,(O R)/(O C)=1/3 and (O S)/(A B)=lambdadot (A) lambda=1/2 (B) lambda=-1 (C) lambda=0 (D) for no value of lambda

A B C D is a parallelogram whose diagonals intersect at Odot If P is any point on B O , prove that: a r\ ( A D O)=A R\ ( C D O) a r\ (\ A B P)=\ a r\ (\ C B P)

P is a point on positive x-axis, Q is a point on the positive y-axis and ' O ' is the origin. If the line passing through P\ a n d\ Q is tangent to the curve y=3-x^2 then find the minimum area of the triangle O P Q , is 3 (b) 4 (c) 5 (c) 6

The side A B of a parallelogram A B C D is produced to any point Pdot A line through A parallel to C P meets C B produced in Q and the parallelogram P B Q R completed. Show that a r(^(gm)A B C D)=a r(^(gm)B P R Q)dot CONSTRUCTION : Join A C and PQ. TO PROVE : a r(^(gm)A B C D)=a r(^(gm)B P R Q)