Prove that
`|(a,b,ax+by),(b,c,bx+cy),(ax+by, bx + cy, 0)| = (b^(2)-ac)(ax^(2) + 2bxy + cy^(2))`.

Prove that |(-a^(2),ab,ac),(ab,-b^(2),bc),(ac,bc,-c^(2))| = 4a^(2)b^(2)c^(2) .

Using the properties of determinant, prove that |(a^(2) +1, ab, ac),(ab, b^(2) + 1, bc),(ac, bc, c^(2)+1)| = 1+a^(2) + b^(2) + c^(2) .

Solve: a(x+b) = 2ax + b

Find dy/dx : y = e^(ax^2 + bx+c)

Using properties of determinant show that : |(a^2,bc,c^2+ac),(a^2+ab,b^2,ac),(ab,b^2+bc,c^2)|=4a^2b^2c^2

Prove that (a^2+2bc)/((a-b)(a-c))+(b^2+2ac)/((b-c)(b-a))+(c^2+2ab)/((c-a)(c-b))=3

Solve by any method: ax+by=c, bx+ay=1+c

If a + b + c = 0 , then prove that (a+b)^2/(ab) + (b+c)^2/(bc) + (c+a)^2/(ca)=3

In a triangle ABC, prove that (b^2 - c^2) cot A + (c^2 -a^2) cot B+ (a^2-b^2) cot C = 0

Solve by the method of substitution: ax+by=1, bx+ay=(a+b)^2/(a^2+b^2)-1