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.

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|

Show that |a b c a+2x b+2y c+2z x y z|=0

If x=b-c+a, y=c-a+b, z=a-b+c , then prove that (b-a) x + (c-b)y +(a-c)z=0

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

If b gt a , d gt c then are of quadrilateral formed by straight lines x = b, x = a, y = c and y =d is

The product of all values of t , for which the system of equations (a-t)x+b y+c z=0,b x+(c-t)y+a z=0,c x+a y+(b-t)z=0 has non-trivial solution, is |a-c-b-c b-a-b-a c| (b) |a b c b c a c a b| |a c bb a cc b a| (d) |a a+bb+c bb+cc+a cc+a a+b|

The determinant |y^2-x y x^2a 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|