If the points ` (a,b),(a_1,b_10 and (a-a_1,b-b_1)` are collinear, show that `a/a_1=b/b_1`

If the points (a, b), (a_1, b_1) and (a-a_1 , b-b) are collinear, show that a/a_1 = b/b_1

If the points (a_1, b_1),\ \ (a_2, b_2) and (a_1+a_2,\ b_1+b_2) are collinear, show that a_1b_2=a_2b_1 .

If (b_2-b_1)(b_3-b-1)+(a_2-a_1)(a_3-a_1)=0 , then prove that the circumcenter of the triangle having vertices (a_1,b_1),(a_2,b_2) and (a_3,b_3) is ((a_2+a_3)/(2),(b_2+b_3)/(2)) .

If the equation of the locus of a point equidistant from the points (a_1,b_1) and (a_2,b_2) is (a_1-a_2)x+(b_2+b_2)y+c=0 , then the value of C is

If ax^2+2bx+c=0 and a_1x^2+2b_1x+c_1=0 have commonroot and a/a_1,b/b_1,c/c_1 are in A.P., show that are: ax^2+2bx+c=0

If |((a_1 - a)^2,(a_1 - b)^2,(a_1 - c)^2),((b_1 - a)^2,(b_1 - b)^2,(b_1 - c)^2),((c_1 - a)^2,(c_1-b)^2,(c_1-c)^2)|=0 and if the vectors vec alpha = (1, a, a^2), vec beta = (1,b,b^2),vec gamma=(1,c,c^2) are non coplanar, show that the vectors vec alpha_1 =(1,a,a_1^2) ,vec beta_1=(1,b_1,b_1^2),vec gamma_1=(1,c_1,c_1^2) are coplanar.

In direct proportion a_1/b_1 = a_2/b_2

Let bi gt 1 for i = 1, 2, ...., 101 . Suppose log_e b_1, log_e b_2, ...., log_e b_101 are in Arithmetic Progression (A.P.) with the common difference log_e 2 . Suppose a_1, a_2, ....., a_101 are in A.P. such that a_1 = b_1 and a_51 = b_51 . If t = b_1 + b_2 + .... + b_51 and s = a_1 + a_2 + .... + a_51 , then .

If the tangent and normal to xy=c^2 at a given point on it cut off intercepts a_1, a_2 on one axis and b_1, b_2 on the other axis, then a_1 a_2 + b_1 b_2 = (A) -1 (B) 1 (C) 0 (D) a_1 a_2 b_1 b_2