Statement 1 : Points `A(1,0),B(2,3),C(5,3),a n dD(6,0)` are concyclic. Statement 2 : Points `A , B , C ,a n dD` form an isosceles trapezium or `A Ba n dC D` meet at `Edot` Then `E A. E B=E C.E D dot`

Four points A(6,3),B(-3,5),C(4,-2) and D(x ,2x) are given in such a way that ((A r e aofD B C))/((A r e aofA B C))=1/2dot

Statement 1:Let A( vec a),B( vec b)a n dC( vec c) be three points such that vec a=2 hat i+ hat k , vec b=3 hat i- hat j+3 hat ka n d vec c=- hat i+7 hat j-5 hat kdot Then O A B C is a tetrahedron. Statement 2: Let A( vec a),B( vec b)a n dC( vec c) be three points such that vectors vec a , vec ba n d vec c are non-coplanar. Then O A B C is a tetrahedron where O is the origin.

If D ,Ea n dF are three points on the sides B C ,C Aa n dA B , respectively, of a triangle A B C such that the (B D)/(C D),(C E)/(A E),(A F)/(B F)=-1

Given three points are A(-3,-2,0),B(3,-3,1)a n dC(5,0,2)dot Then find a vector having the same direction as that of vec A B and magnitude equal to | vec A C|dot

Statement 1: Let vec a , vec b , vec ca n d vec d be the position vectors of four points A ,B ,Ca n dD and 3 vec a-2 vec b+5 vec c-6 vec d=0. Then points A ,B ,C ,a n dD are coplanar. Statement 2: Three non-zero, linearly dependent coinitial vector ( vec P Q , vec P Ra n d vec P S) are coplanar. Then vec P Q=lambda vec P R+mu vec P S ,w h e r elambdaa n dmu are scalars.

Statement 1: Let vec a , vec b , vec ca n d vec d be the position vectors of four points A ,B ,Ca n dD and 3 vec a-2 vec b+5 vec c-6 vec d=0. Then points A ,B ,C ,a n dD are coplanar. Statement 2: Three non-zero, linearly dependent coinitial vector ( vec P Q , vec P Ra n d vec P S) are coplanar. Then vec P Q=lambda vec P R+mu vec P S ,w h e r elambdaa n dmu are scalars.

In A B C , on the side B C ,Da n dE are two points such that B D=D E=E Cdot Also /_A D E=/_A E D=alpha, then

Statement 1: Distance of point D( 1,0,-1) from the plane of points A( 1,-2,0) , B ( 3, 1,2) and C( -1,1,-1) is 8/sqrt229 Statement 2: volume of tetrahedron formed by the points A,B, C and D is sqrt229/ 2

Statement 1: if three points P ,Qa n dR have position vectors vec a , vec b ,a n d vec c , respectively, and 2 vec a+3 vec b-5 vec c=0, then the points P ,Q ,a n dR must be collinear. Statement 2: If for three points A ,B ,a n dC , vec A B=lambda vec A C , then points A ,B ,a n dC must be collinear.

Two circles C_1a n dC_2 both pass through the points A(1,2)a n dE(2,1) and touch the line 4x-2y=9 at Ba n dD , respectively. The possible coordinates of a point C , such that the quadrilateral A B C D is a parallelogram, are (a ,b)dot Then the value of |a b| is_________