Prove that
`|{:(x_1,y_1,1),(ax_1+bx_2+cx_3,ay_1+by_2+cy_3,a+b+c),(-ax_1+bx_2+cx_3,-ay_1+by_2+cy_2,-a+b+c):}|=0`

if (x_(1)-x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2). where a,b,c are positive then prove that 4 |{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}| = (a+b+c) (b+c-a) (c+a-b)(a+b-c)

Prove that: |[a, b, ax+by],[ b, c, bx+cy], [ax+by, bx+cy,0]|=(b^2-a c)(a x^2+2b x y+c y^2)

Show that the plane ax+by+cz+d=0 divides the line joining (x_1, y_1, z_1) and (x_2, y_2, z_2) in the ratio of (-(ax_1+ay_1+cz_1+d)/(ax_2+by_2+cz_2+d))

ax + by = c bx+ ay =1 +c

If a,b,c epsilon R, a!=0 and (b-1)^2lt4ac then the number of real roots of the system equations (in three unknowns x_1,x_2,x_3) ax_1^2+bx_1+c=x_2 , ax_2^2+bx_2+c=x_3 , ax_3^2+bx_3+c=x_1 is (A) 0 (B) 1 (C) 2 (D) 3

If |{:(1+ax,1+bx,1+cx),(1+a_1x,1+b_1x,1+c_1x),(1+a_2x,1+b_2x,1+c_2x):}|=A_0+A_1x+A_2x^2+A ""_3x^3 , then A_0 is

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) .

Factorise : ax + 2bx + 3cx - 3a - 6b - 9c

{:(ax + by = 1),(bx + ay = ((a + b)^(2))/(a^(2) + b^(2))-1):}

If y =(ax +b)/( cx + d) , then prove that 2y_1y_3=3(y_2)^2