`A(1, -2) and B(2, 5)` are two points. The lines `OA, OB` are produced to `C and D` respectively, such that `OC=2OA and OD=2OB`. Find `CD`.

On the xy plane where O is the origin, given points, A(1, 0), B(0, 1) and C(1, 1) . Let P, Q, and R be moving points on the line OA, OB, OC respectively such that overline(OP)=45t overline((OA)),overline(OQ)=60t overline((OB)),overline(OR)=(1-t) overline((OC)) with t>0. If the three points P,Q and R are collinear then the value of t is equal to A (1)/(106) B (7)/(187) C (1)/(100) D none of these

O is the origin and lines OA, OB and OC have direction cosines l_(r),m_(r)andn_(r)(r=1,2 and3) . If lines OA', OB' and OC' bisect angles BOC, COA and AOB, respectively, prove that planes AOA', BOB' and COC' pass through the line (x)/(l_(1)+l_(2)+l_(3))=(y)/(m_(1)+m_(2)+m_(3))=(z)/(n_(1)+n_(2)+n_(3)) .

If D , E and F are the mid-points of the sides BC , CA and AB respectively of the DeltaABC and O be any point, then prove that OA+OB+OC=OD+OE+OF

Two concentric shells of uniform density of mass M_(1) and M_(2) are situated. The forces experienced by a particle of mass m when palced at positions A,B and C respectively are given OA =p,OB = q and OC =r . .

A line through the origin intersects x = 1,y =2 and x +y = 4 , in A, B and C respectively,such that OA*OB*OC =8 sqrt2 . Find the equation of the line.

The following figure shows a trapezium ABCD in which AB is parallel to DC and AD= BC Prove that: OA = OB and OC= OD

Two particles 1 and 2 are allowed to descend on the two frictionless chord OA and OB of a vertical circle, at the same instant from point O . The ratio of the velocities of the particles 1 and 2 respectively, when they reach on the circumference will be (OB is the diameter).

In the figure, OA and OB are two uniformly charged wires, placed along x and y axis respectively. Find the unit vector in the direction of field at point (0,0,sqrt(3L)) .

Two fixed point A and B are taken on the two axes such that OA=a, OB=b , two variable points C, D are taken on the same axes respectively, find the locus of the point of intersection of AD and CB if : 1/OC - 1/OD = 1/OA - 1/OB

A and B are two points on the parabola y^(2) = 4ax with vertex O. if OA is perpendicular to OB and they have lengths r_(1) and r_(2) respectively, then the valye of (r_(1)^(4//3)r_(2)^(4//3))/(r_(1)^(2//3)+r_(2)^(2//3)) is