Factorize: `a x+b x+a y+b y` `+a x^2+b y^2+b x^2+a y^2` `a^2+b c+a b+a c`+ `a x-a y+b x-b y`

Prove that |[a x-b y-c z, a y+b x, c x+a z], [a y+b x, b y-c z-a x, b z+c y],[c x+a z, b z+c y, c z-a x-b y]|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

The image of the pair of lines represented by a x^2+2h x y+b y^2=0 by the line mirror y=0 is a. a x^2-2h x y-b y^2=0 b. b x^2-2h x y+a y^2=0 c. x^2+2h x y+a y^2=0 d. a x^2-2h x y+b y^2=0

The determinant |[y^2,-x y, x^2],[a, b, c],[ a ', b ', c ']| is equal to a. |[b x+a y, c x+b y],[ b^(prime)x+a ' y, c^(prime)x+b ' y]| b. |[a x+b y, b x+c y],[ a^(prime)x+b ' y, b ' x+c ' y]| c. |[b x+c y, a x+b y],[ b^(prime)x+c ' y, a^(prime)x+b ' y]| d. |[a x+b y, b x+c y],[ a^(prime)x+b ' y, b^(prime)x+c ' y]|

Let a ,b ,c be real numbers with a^2+b^2+c^2=1. Show that the equation |a x-b y-c b x-a y c x+a b x+a y-a x+b y-cc y+b c x+a c y+b-a x-b y+c|=0 represents a straight line.

Let's find value of : (x - y)a + (y - z) b + (z -x)c if a = x^2 + xy + y^2,b = y^2 + yz + z^2, c = z^2 + zx + x^2

Prove that (a b+x y)(a x+b y)>4a b x y(a , b ,x ,y >0)dot

If a^2+b^2+c^2=x^2+y^2+z^2=1, then show that a x+b y+c z leq1.

If x : a = y : b = z : c , then prove that (a^(2) + b^(2) + c^(2))(x^(2) + y^(2) + z^(2)) = (ax + by + cz)^(2)

Find the condition that the expressions a x^2-b x y+c y^2a n da_1x^2+b_1x y+c_1y^2 may have factors y-m xa n dm y-x , respectively.

If x+y+z=0 prove that |[a x, b y, c z],[ c y, a z ,b x],[ b z ,c x ,a y]|=x y z|[a, b, c],[c ,a ,b],[b ,c ,a]|