Area of the triangle with vertices `(a, b), (x_1,y_1) and (x_2, y_2)` where `a, x_1,x_2` which in G.P. common ratiorand `b, y_1, y_2`, are in G.P with common ratio s, is

The area of the triangle OAB with vertices O(0,0),A(x_(1),y_(1))B(x_(2),y_(2)) is

The area of the triangle OAB with vertices O(0,0),A(x_(1),y_(1))" and B(x_(2),y_(2)) is

The centroid of the triangle with vertices at A(x_(1),y_(1)),B(x_(2),y_(2)),c(x_(3),y_(3)) is

" The area of the triangle OAB with vertices "O(0,0),A(x_(1),y_(1))" and "B(x_(2),y_(2))" is "ul(sq)" units "

P(x, y) moves such that the area of the triangle with vertices at P(x, y), (1, -2), (-1, 3) is equal to the area of the triangle with vertices at P(x, y), (2, -1), (3, 1). The locus of P is

We know that, if a_1, a_2, ..., a_n are in H.P. then 1/a_1,1/a_2,.....,1/a_n are in A.P. and vice versa. If a_1, a_2, ..., a_n are in A.P. with common difference d, then for any b (>0), the numbers b^(a_1),b^(a_2),b^9a_3),........,b^(a_n) are in G.P. with common ratio b^d. If a_1, a_2, ..., a_n are positive and in G.P. with common ration, then for any base b (b> 0), log_b a_1 , log_b a_2,...., log_b a_n are in A.P. with common difference logor.If x, y, z are respectively the pth, qth and the rth terms of an A.P., as well as of a G.P., then x^(z-x),z^(x-y) is equal to

If a, x_(1), x_(2) are in G.P. with common ration r, and b, y_(1), y_(2) are in G.P. with common ratio s where s-r=2 , then the area of the triangle with vertices (a, b), (x_(1), y_(1)) and (x_(2),y_(2)) is