In a circle, chords `A Ba n dC D` intersect at a point `R` inside the circle. If `A R : R B=1:4a n dC R : R D=4:9,` then the ratio `A B : C D` is_______.

In a circle, chords AB and CD intersect at a point R inside the circle. If AR : RB = 1:4 and CR: RD = 4:9, then the ratio AB: CD is

In a circle with centre O , chords A Ba n dC D intersect inside the circumference at Edot Prove that /_A O C+/_B O D=2/_A E Cdot

Let the base A B of a triangle A B C be fixed and the vertex C lies on a fixed circle of radius rdot Lines through Aa n dB are drawn to intersect C Ba n dC A , respectively, at Ea n dF such that C E: E B=1:2a n dC F : F A=1:2 . If the point of intersection P of these lines lies on the median through A B for all positions of A B , then the locus of P is a circle of radius r/2 a circle of radius 2r a parabola of latus rectum 4r a rectangular hyperbola

A B C D is a parallelogram whose diagonals A C\ a n d\ B D intersect at O . A line through O intersects A B\ a t\ P\ a n d\ D C\ at Q . Prove that a r\ (\ /_\P O A)=\ a r\ (\ /_\Q O C) .

Diagonals A C\ a n d\ B D of a quadrilateral A B C D intersect at O in such a way that a r\ ( A O D)=a r\ (\ B O C)dot Prove that A B C D is a trapezium.

A D ,\ B E\ A N D\ C F are the perpendiculars from the angular points of a A B C upon the opposite sides. The perimeters of the D E F\ a n d\ A B C are in the ratio. (2r)/R (b) r/(2R) (c) r/R (d) r/(3R) Where r is the inradius and R is circum-radius of the A B C

A point O inside a rectangle A B C D is joined to the vertices. Prove that the sum of the areas of a pair of opposite triangles so formed is equal to the sum of the other pair of triangles. Given: A rectangle A B C D\ a n d\ O is a point inside it. O A ,\ O B ,\ O C\ a n d\ O D have been joined. To Prove: a r\ (A O D)+\ a r\ ( B O C)=\ a r\ ( A O B)+\ a r( C O D)

If inside a big circle exactly n(nlt=3) small circles, each of radius r , can be drawn in such a way that each small circle touches the big circle and also touches both its adjacent small circles, then the radius of big circle is r(1+cos e cpi/n) (b) ((1+tanpi/n)/(cospi/pi)) r[1+cos e c(2pi)/n] (d) (r[s inpi/(2n)+cos(2pi)/n]^2)/(sinpi/n)

The diagonals of quadrilateral A B C D intersect at O . Prove that (a r( A C B))/(a r( A C D))=(B O)/(D O)

Statement 1: If the endpoints of two normal chords A Ba n dC D (normal at Aa n dC) of a parabola y^2=4a x are concyclic, then the tangents at Aa n dC will intersect on the axis of the parabola. Statement 2: If four points on the parabola y^2=4a x are concyclic, then the sum of their ordinates is zero.