The most general form of a quadratic is: $ax^2 + by^2 + 2gx + 2fy + c + 2hxy= 0 $ and that for a homogeneous second degree equation is : $ax^2 + by^2 + 2hxy=0$
I derived the formula $$\tan \theta = \left|\dfrac{2\sqrt{h^2 – ab}}{a+b}\right|$$
but later, author claimed that the same formula is applicable for the general form too.
But I (and he too) derived it for the homogeneous case only. How can he claim this then?
Best Answer
One can write $$0 = ax^2+by^2+2gx+2fy+c+2hxy = (a_1x+b_1y+c_1)(a_2x+b_2y+c_1)$$
So, the angle between two lines is $$\tan \theta = \left|\frac{\tan \alpha -\tan \beta}{1+\tan\alpha\tan\beta}\right| = \left|\frac{-\frac{a_1}{b_1}+\frac{a_2}{b_2}}{1+\frac{a_1a_2}{b_1b_2}}\right| = \left|\frac{-a_1b_2+b_1a_2}{a_1a_2+b_1b_2}\right|= \left|\frac{\sqrt{(a_1b_2+a_2b_1)^2-4a_1a_2b_1b_2}}{a+b}\right| = \left|\frac{\sqrt{4h^2-4ab}}{a+b}\right|.$$