Given curves: $$ax^2+2hxy+by^2+2gx=0$$ $$a_1x^2+2h_1xy+b_1y^2+2g_1x=0$$
It is given that the lines joining the origin to the points of intersection of these curves are perpendicular to each other.
We've to prove that: $$\boxed{g(a_1+b_1)=g_1(a+b)}$$ using homogenisation
My approach
Let $lx+my=n$ be the line passing through the points of intersection. $$\frac{lx+my}{n}=1$$
Now, transforming the curves into second degree homogenous equations, we get: $$ax^2+2hxy+by^2+2gx\frac{lx+my}{n}=0$$ $$a_1x^2+2h_1xy+b_1y^2+2g_1x\frac{lx+my}{n}=0$$
These are the required lines.
How to proceed from here?