Why is $h^2-ab$ the discriminant of the conic with equation $ax^2+2hxy+by^2+2gx+2fy+c=0$?

Question

I recently began my study of conic sections at high school. the term $h^2-ab$ was declared the discriminant of equation $$ax^2+2hxy+by^2+2gx+2fy+c=0$$ (which represents a conic or pair of lines). I tried my best to derive the given discriminant from the given standard equation of any conic but all in vain. A good reasoning behind why $h^2-ab$ is the discriminant of the standard equation of any conic would be appreciated.

Answer 1

The following development is based on the given equation represents a pair of straight line only.

By suitable translation, the given pair of straight lines is pair-wisely parallel to $$ax^2 + 2hxy + by^2= 0$$ Equivalently, we have a reduced equation:

$$(\dfrac yx)^2 + \dfrac {2h}{b}(\dfrac {y}{x}) + \dfrac {a}{b} = 0$$

If it represents a pair of straight lines, then we should have a pair of straight line passing through the origin and hence $$(\dfrac {y}{x} – m_1)( \dfrac {y}{x} – m_2) = 0$$ where $m_1$ and $m_2$ are the slopes of those two lines.

On the other hand, $m_1$, and $m_2$ are the roots of the reduced equation. Then, $m_1 + m_2 = ….$ and $m_1m_2 = ….$.

Note that if $\theta$ is the angle between these two lines, then

$$\tan \theta = (+/-) \dfrac {m_1 – m_2}{1 + m_1m_2} = … = (+/-) \dfrac {2\sqrt {h^2 – ab}}{a + b}$$

The values a, b and h can affect $h^2 – ab$. That, in turn, can be used to determine $\theta$ and hence the condition of those two lines.