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

Prove that: |a b a x+b y b c b x+c y a x+b y b x+c y0|=(b^2-a c)(a x^2+2b x y+c y^2) .

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

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

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

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

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]|

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]|

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 cc a bb c a|

A circle with center (a , b) passes through the origin. The equation of the tangent to the circle at the origin is (a) a x-b y=0 (b) a x+b y=0 b x-a y=0 (d) b x+a y=0

Prove that |(x+2a,y+2b,z+2c),(x,y,z),(a,b,c)|=0