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` .

Prove that the points (a ,\ b),\ (a_1,\ b_1) and (a-a_1,\ b-b_1) are collinear if a b_1=a_1b .

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_1-b_2)y+c=0 , then the value of c is a a2-a2 2+b1 2-b2 2 sqrt(a1 2+b1 2-a2 2-b2 2) 1/2(a1 2+a2 2+b1 2+b2 2) 1/2(a2 2+b2 2-a1 2-b1 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_1-b_2)y+c=0 , then find the value of c .

If the lines a_1x+b_1y+1=0,\ a_2x+b_2y+1=0\ a n d\ a_3x+b_3y+1=0 are concurrent, show that the points (a_1, b_1),\ (a_2, b_2)a n d\ (a_3, b_3) are collinear.

If the lines a_1x+b_1y+1=0,\ a_2x+b_2y+1=0\ a n d\ a_3x+b_3y+1=0 are concurrent, show that the points (a_1, b_1),\ (a_2, b_2)a n d\ (a_3, b_3) are collinear.

The ratio in which the line segment joining points A(a_1,\ b_1) and B(a_2,\ b_2) is divided by y-axis is (a) -a_1: a_2 (b) a_1: a_2 (c) b_1: b_2 (d) -b_1: b_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 the origin lies in the acute angle between the lines a_1 x + b_1 y + c_1 = 0 and a_2 x + b_2 y + c_2 = 0 , then show that (a_1 a_2 + b_1 b_2) c_1 c_2 lt0 .

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)) .

Straight lines y=m x+c_1 and y=m x+c_2 where m in R^+, meet the x-axis at A_1a n dA_2, respectively, and the y-axis at B_1a n dB_2, respectively. It is given that points A_1,A_2,B_1, and B_2 are concylic. Find the locus of the intersection of lines A_1B_2 and A_2B_1 .