Prove that: `|(1, 1, 1),( 1, 1+x,1) ,(1, 1, 1+y)|=x y` .

If D=|(1, 1, 1), (1, 1+x, 1), (1, 1, 1+y)| for x!=0, y!=0 then D is (1) divisible by neither x nor y (2) divisible by both x and y (3) divisible by x but not y (4) divisible by y but not x

Prove that : |{:(1,x,yz),(1,y,zx),(1,z,xy):}|=(x-y)(y-z)(z-x)

If A(x_1, y_1), B(x_2, y_2), C(x_3, y_3) are the vertices of a DeltaABC and (x, y) be a point on the median through A . Show that : |(x, y, 1), (x_1, y_1, 1), (x_2, y_2, 1)| + |(x, y, 1), (x_1, y_1, 1), (x_3, y_3, 1)|=0

Without expanding prove that: |[x+y, y+z, z+x],[z ,x, y],[1, 1 ,1]|=0 .

Without expanding, prove that Delta=|(x+y,y+z,z+x),(z,x,y),(1,1,1)|=0

Prove that : =|{:(1,1,1),(x,y,z),(x^(2),y^(2),z^(2)):}|=(x-y)(y-z)(z-x)

If A(x_(1), y_(1)), B(x_(2), y_(2)) and C (x_(3), y_(3)) are the vertices of a Delta ABC and (x, y) be a point on the internal bisector of angle A, then prove that b|(x,y,1),(x_(1),y_(1),1),(x_(2),y_(2),1)|+c|(x,y,1),(x_(1),y_(1),1),(x_(3),y_(3),1)|=0 where, AC = b and AB = c.

Three points A(x_1 , y_1), B (x_2, y_2) and C(x, y) are collinear. Prove that: (x-x_1) (y_2 - y_1) = (x_2 - x_1) (y-y_1) .

Using properties of determinats prove that : |(x,x(x^(2)+1),x+1),(y,y(y^(2)+1),y+1),(z,z(z^(2)+1),z +1)|=(x-y)(y-z)(z-x)(x+y+z)

Tangents are drawn from the points (x_(1), y_(1))" and " (x_(2), y_(2)) to the rectanguler hyperbola xy = c^(2) . The normals at the points of contact meet at the point (h, k) . Prove that h [1/x_(1) + 1/x_(2)] = k [1/y_(1)+ 1/y_(2)] .