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.

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

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

Let a and b be nonzero real numbers. Then the equation (a x^2+b y^2+c)(x^2-5x y+6y^2)=0 represents. four straight lines, when c=0 and a , b are of the same sign. two straight lines and a circle, when a=b and c is of sign opposite to that a two straight lines and a hyperbola, when aa n db are of the same sign and c is of sign opposite to that of a a circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a

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

Equations (b-c)x+(c-a)y+(a-b)=and(b^(3)-c^(3))x+(c^(3)-a^(3))y+a^(3)-b^(3)=0 will represent the same straight line if -

Show that the reflection of the line a x+b y+c=0 on the line x+y+1=0 is the line bx+a y+(a+b-c)=0 where a!=bdot

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

Show that the straight lines x/a+y/b=1/c, x/b+y/c=1/a and x/c+y/a=1/b will be concurrent if ab+bc+ca=0

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 a, b, c are in GP and a, x, b, y are in AP Then prove that, a/x + c/y = 2