Straight lines

i don't think this is in syllabus

This is a case of rotation of axes without shifting the origin
S=ax + 2hxy + by
Let the axes be rotated through θ
then x=x^'cosθ + y^'sinθ
and y=x^'sinθ + y^'cosθ
where (x^',y^') are new coordinates
so S'=a(x^'cosθ + y^'sinθ) +2h(x^'cosθ + y^'sinθ)(x^'sinθ + y^'cosθ) +b(x^'sinθ + y^'cosθ)

Now a'=coff. of x'²=a cos²Î¸+2hsinθcosθ+bsin²Î¸
h'=(1/2)coff. of x'y' =-a sinθcosθ+h cos²Î¸-h sin²Î¸+b sinθcosθ
b'=coff. of y'² =a sin²Î¸-2hsinθcosθ+bcos²Î¸

Now a'+b'=a(cos²Î¸+sin²Î¸) +b(cos²Î¸+sin²Î¸)
=a+b

Similarly u can prove h'²-a'b'=h²-ab