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,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_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 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 |((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.

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

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 (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)) .

Let A={a_1,a_2,a_3,.....} and B={b_1,b_2,b_3,.....} are arithmetic sequences such that a_1=b_1!=0, a_2=2b_2 and sum_(i=1)^10 a_i=sum_(i=1)^15 b_j , If (a_2-a_1)/(b_2-b_1)=p/q where p and q are coprime then find the value of (p+q).

Let b_i > 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 a,b are positive quantitiesand if a_1= (a+b)/2, b_1= sqrt(a_1b,) a_2= (a_1+b_1)/2, b_2= sqrt(a_2b_1 and so on show that Lt_(nrarroo) a_n =Lt_(nrarroo) b_n = (sqrt(b^2-a^2))/(cos^-1 (a/b))