`C_1` and `C_2` are two concentric circles, the radius of `C_2` being twice that of `C_1`. From a point P on `C_2`, tangents PA and PB are drawn to `C_1`. Prove that the centroid of the triangle PAB lies on `C_1` .

The centres of two circles C_(1) and C_(2) each of unit radius are at a distance of 6 unit from each other. Let P be the mid-point of the line segment joining the centres of C_(1) and C_(2) and C be a circle touching circles C_(1) and C_(2) externally. If a common tangent to C_(1) and C passing through P is also a common tangent to C_(2) and C, then the radius of the circle C, is

A straight line intersects thethree concentric circles at A, B, C. if the distance of line from the centre of the circles is 'P', prove that the area of the triangle formed by tangents to the circle at A, B, C is ((1)/(2P).AB.BC.CA) .

Let C_1 and C_2 be two circles with C_2 lying inside C_1 A circle C lying inside C_1 touches C_1 internally and C_2 externally. Identify the locus of the centre of C

Let C_1 and C_2 be respectively, the parabolas x^2=y-1 and y^2=x-1 Let P be any point on C_1 and Q be any point on C_2 . Let P_1 and Q_1 be the refelections of P and Q, respectively with respect to the line y=x. Arithemetic mean of PP_1 and Q Q_1 is always less than

Two planes P_1 and P_2 pass through origin. Two lines L_1 and L_2 also passing through origin are such that L_1 lies on P_1 but not on P_2 , L_2 lies on P_2 but not on P_1 . A,B, C are there points other than origin, then prove that the permutation [A', B', C'] of [A, B, C] exists. Such that: A lies on L_1 , B lies on P_1 not on L_1 , C does not lie on P_1 .

Two planes P_1 and P_2 pass through origin. Two lines L_1 and L_2 also passingthrough origin are such that L_1 lies on P_1 but not on P_2 , L_2 lies on P_2 but not on P_1 . A,B, C are there points other than origin, then prove that the permutation [A', B', C'] of [A, B, C] exists. Such that: A' lies on L_2 , B' lies on P2 not on L_2 , C' does not lies on P_2 .

Prove that the points A (1,2,3), B(2,3,1) and C(3,1,2) are the vertices of an equilateral triangle.

Find the equation of plane passing through the points : (2, - 1, 1), B (4, 3, 2) and C (6, 5, -2). Also, prove. that the point (5, -1,-2:) lies on the plane given by A, B and C.

Statement I If sum of algebraic distances from points A(1,2),B(2,3),C(6,1) is zero on the line ax+by+c = 0 then 2a+3b + c = 0 , Statement II The centroid of the triangle is (3,2)

If C_(1): x^(2)+y^(2) =(3+2sqrt(2))^(2) be a circle. PA and PB are pair of tangents on C_(1) where P is any point on the director circle of C_(1) , then the radius of the smallest circle which touches C_(1) externally and also the two tangents PA and PB is