Question

The condition for the equation
ax^2 +2hxy+by^2 + 2 gx + 2 fy+c=0 can take the
form ax^2 +2hxy+by^2 =0 by translation of
origin is_______
( please answer this question)​

Answer 1

ANSWER

The general solution is ax2+2hxy+by2+2gx+2fy+c=0  ....(1)

Let (α,β) be the point of intersection we consider line paralleled transformation.

Let x=x′+α,y=y′+β

From (1) we have

a(x′+α)2+2h(x′+α)(y′+β)+b(y′+β)2+2g(x′+α)+2f(y′+β)+c=0

⇒ax′2+2hx′y′+by′2+aα2+2hαβ+bβ2+2gα+2fβ+2x′(aα+hβ+g)+2y′+2y′(hα+bβ+f)=0

⇒ax′