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`.

(1) The straight lines (2k+3) x + (2-k) y+3=0 , where k is a variable, pass through the fixed point (-3/7, -6/7) . (2) The family of lines a_1 x+ b_1 y+ c_1 + k (a_2 x + b_2 y + c_2) = 0 , where k is a variable, passes through the point of intersection of lines a_1 x + b_1 y + c_1 = 0 and a_2 x+ b_2 y + c_2 = 0 (A) Both 1 and 2 are true and 2 is the correct explanation of 1 (B) Both 1 and 2 are true and 2 is not a correct explanation of 1 (C) 1 is true but 2 is false (D) 1 is false but 2 is true

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 one of the roots of the equation ax^2 + bx + c = 0 be reciprocal of one of the a_1 x^2 + b_1 x + c_1 = 0 , then prove that (a a_1-c c_1)^2 =(bc_1-ab_1) (b_1c-a_1 b) .

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

Show that the circle passing through the origin and cutting the circles x^2+y^2-2a_1x-2b_1y+c_1=0 and x^2+y^2-2a_2x -2b_2y+c_2=0 orthogonally is |(x^2+y^2,x,y),(c_1,a_1,b_1),(c_2,a_2,b_2)|=0

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

If G_1 is the mean centre of A_1,B_1,C_1 and G_2 that of A_2,B_2,C_2 then show that vec(A_1A_2)+vec(B_1B_2)+vec(C_1C_2)=3vec(G_1G_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 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 A_1 and A_2 are two AMs between a and b then (2A_1 -A_2) (2A_2 - A_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