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In one of the books I referred for conic sections, it gave a statement

If a conic represents an empty set, then $\Delta \neq 0$ and $h^2<ab$.

Can someone please explain this statement..? What is the meaning of a conic representing an empty set, actually..?

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user821928
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    What is $\Delta$? What is $h$? What is $a$? What is $b$? Please give some more details to clarify the problem. What do you mean by "empty"? A conic section is a set of points in $Bbb{R}^2$ and is thus never empty. – K.defaoite Nov 09 '20 at 11:34
  • I have revised my answer, you will be happy to see but be careful. – Z Ahmed Nov 09 '20 at 14:59

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If $a,b,c,f,g,h$ are chosen randomly, then the chance that the quadratic (1) will represent a null set is the highest. There is no unique parametric condition. See three cases below

The equation of conic/ pair of lines is given by $$ax^2+by^2+2hxy+2gx+2fy+c=0~~~~(1)$$ See it as a quadratic of $x$ and treat $y$ as constant, then $$x=\frac{-(hy+g)\pm \sqrt{(h^2-ab)y^2+2(gh-af)y+g^2-ac}}{a}~~~~(2)$$ Eq. (1) will represent null set if the discriminant is negative drfinite (for all real values of $y$). $f(x)=Ax^2+Bx+C \le 0 ~\forall~ y\in R ~~if~~ A<0~~and ~~ \Delta=B^2-4AC<0$. This implies $$h^2<ab~~~(3)$$ and $$\Delta=(gh-af)^2-(h^2-ab)(g^2-ac)=abc+2fgh-af^2-bg^2-ch^2 <0~~~~(4).$$ (3) and (4) note $\Delta <0$ not ($\Delta \ne 0).$ (Case 1).

When $h^2=ab, gh=af$ but $g^2<ac$, we will have $\Delta=0$ so two parallel (imaginary)lines (2). This also corresponds to a null set. (Case 2)

Let $g^2-ac=0, gh-af=0, h^2<ab$, then you get $\Delta=2fgh-af^2-ch^2$ may be positive or negative. One again gets two (imaginary) lines (1). (Case 3)

Case 3, justifies OP's obervation in a book.

Z Ahmed
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