ABCD is a tetrahedron and O is any point. If the lines joining O to the vertices meet the opposite faces at `P, Q, R and S`, prove that `(OP)/(AP)+ (OQ)/(BQ) + (OR)/(CR) + (OS)/(DS)=1`.

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.

ABCD is a quadrilateral and E is the point of intersection of the lines joining the middle points of opposite side. Show that the resultant of vec (OA) , vec(OB) , vec(OC) and vec(OD) = 4 vec(OE) , where O is any point.

TP and TQ are any two tangents to a parabola and the tangent at a third point R cuts them in P' and Q' . Prove that (TP')/(TP)+(TQ')/(TQ)=1

A triangle A B C is inscribed in a circle with centre at O , The lines A O ,B Oa n dC O meet the opposite sides at D , E ,a n dF , respectively. Prove that 1/(A D)+1/(B E)+1/(C F)=(acosA+bcosB+ccosC)/

A B C D is a quadrilateral. E is the point of intersection of the line joining the midpoints of the opposite sides. If O is any point and vec O A+ vec O B+ vec O C+ vec O D=x vec O E ,t h e nx is equal to a. 3 b. 9 c. 7 d. 4

ABCD is quadrilateral with AB parallel to DC. A line drawn parallel to AB meets AD at P and BC at Q. prove that (AP)/(PD) = (BQ)/(QC)

Show that the angle between the tangent at any point P and the line joining P to the origin O is same at all points on the curve log(x^2+y^2)=ktan^(-1)(y/x)

In what ratio does the point Q(1,6) divide the line segment joining the points P(2,7) and R(-2,3).

The tangents to the parabola y^2=4a x at the vertex V and any point P meet at Q . If S is the focus, then prove that S P,S Q , and S V are in GP.

If the normals at P, Q, R of the parabola y^2=4ax meet in O and S be its focus, then prove that .SP . SQ . SR = a . (SO)^2 .