If `l, l_(1), l_(2) and l _(3)` be respectively the radii of the in-circle and the three escribed circles of a `Delta ABC,` then find `l_(1)l_(2) l_(3)-l(l_(1)l_(2) +l_(2) l_(3)+l_(1) l_(3)).`

If l_(1), l_(2), l_(3) are respectively the perpendicular from the vertices of a triangle on the opposite side, then show that l_(1)l_(2) l_(3) =(a^(2)b ^(2) c^(2))/(8R^(3)).

Let L_(1),L_(2),L_(3) be three straight lines a plane and n be the number of circles touching all the lines . Find the value of n.

A=[{:(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)),(l_(3),m_(3),n_(3)):}] and B=[{:(p_(1),q_(1),r_(1)),(p_(2),q_(2),r_(2)),(p_(3),q_(3),r_(3)):}] Where p_(i), q_(i),r_(i) are the co-factors of the elements l_(i), m_(i), n_(i) for i=1,2,3 . If (l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)) and (l_(3),m_(3),n_(3)) are the direction cosines of three mutually perpendicular lines then (p_(1),q_(1), r_(1)),(p_(2),q_(2),r_(2)) and (p_(3),q_(),r_(3)) are

If A = [[l_(1),m_(1),n_(1)],[l_(2),m_(2),n_(2)],[l_(3),m_(3),n_(3)]] then Find A+I

Statement I: If a=2hati+hatk,b=3hatj+4hatk and c=lamda a+mub are coplanar, then c=4a-b . Statement II: A set vector a_(1),a_(2),a_(3), . . ,a_(n) is said to be linearly independent, if every relation of the form l_(1)a_(1)+l_(2)a_(2)+l_(3)a_(3)+ . . .+l_(n)a_(n)=0 implies that l_(1)=l_(2)=l_(3)= . . .=l_(n)=0 (scalar).

Let position vectors of point A,B and C of triangle ABC represents be hati+hatj+2hatk, hati+2hatj+hatk and 2hati+hatj+hatk . Let l_(1),l_(2) and l_(3) be the length of perpendicular drawn from the orthocenter 'O' on the sides AB, BC and CA, then (l_(1)+l_(2)+l_(3)) equals

If l, m, n denote the side of a pedal triangle, then (l)/(a ^(2))+(m)/(b^(2))+(n)/(c ^(2)) is equal to

Let L_(1) -= ax+by+a root3 (b) = 0 and L_(2) -= bx - ay + b root3 (a) = 0 be two straight lines . The equations of the bisectors of the angle formed by the foci whose equations are lambda_(1)L_(1)-lambda_(2)L_(2)=0 and lambda _(1) l_(1) + lambda_(2) = 0 , lambda_(1) and lambda_(2) being non - zero real numbers ,are given by