The equation `(b-c)x+(c-a)y+(a-b)=0` and `(b^3-c^3)x+(c^3-a^3)y+a^3-b^3=0` will represent the same line if

Prove that : |{:(a,b,c),(b,c,a),(c,a,b):}|=3 a b c-a^(3)-b^(3)-c^(3)

Prove that the lines (b-c)x+(c-a)y+(a-b)=0, (c-a)x+(a-b)y+(b-c)=0 and (a-b)x+(b-c)y+(c-a)=0 are concurrent.

If c<1 and the system of equations x+y-1=0,2x-y-c=0,and -bx+3b y-c=0 is consistent, then the possible real values of b are

The equation |(x-a,x-b,x-c),(x-b,x-a,x-c),(x-c,x-b,x-a)|=0 (a,b,c are different) is satisfied by (A) x=(a+b+c0 (B) x= 1/3 (a+b+c) (C) x=0 (D) none of these

Suppose the circle having equation x^2+y^2=3 intersects the rectangular hyperbola x y=1 at points A ,B ,C ,a n dDdot The equation x^2+y^2-3+lambda(x y-1)=0,lambda in R , represents. a pair of lines through the origin for lambda=-3 an ellipse through A ,B ,C ,a n dD for lambda=-3 a parabola through A , B , C ,a n dD for lambda=-3 a circle for any lambda in R

If c<1 and the system of equations x+y-1=0,2x-y-c=0,a n d -b x+3b y-c=0 is consistent, then the possible real values of b are b ∈(-3 3/4) b. b ∈(-3/2,4) c. b∈ (-3/4,3) d. none of these

If a system of three linear equations x+4a y+a=0,x+3b y+b=0, and x+2c y+c=0 is consistent, then prove that a ,b , c are in H.P.

If A(x_1, y_1),B(x_2, y_2),C(x_3, y_3) are the vertices of a triangle, then the equation |x y1x_1y_1 1x_2y_3 1|+|x y1x_1y_1 1x_3y_3 1|=0 represents (a)the median through A (b)the altitude through A (c)the perpendicular bisector of B C (d)the line joining the centroid with a vertex

If the roots of the equation (c^2-a b)x^2-2(a^2-b c)x+b^2-a c=0 are equal, prove that either a=0 or a^3+b^3+c^3=3a b c dot

If the equation (4a-3)x^2+a y^2+6x-2y+2=0 represents a circle, then its centre is a. (3,-1) b. (3,1) c. (-3,1) d. none of these