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In this post, intuition for method of discriminant for determining what kind of conic a second degree equation in $x$ and $y$ is discussed. My question is what would it evaluate for a product of straight lines? Eg: $(y-x)(y+x)=y^2 - x^2$, In this particular example I get $B^2- 4AC=4$ which would suggest a hyperbola but is this generally true?

tryst with freedom
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1 Answers1

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Your specific case involves two orthogonal lines, and the discriminant is positive. That does not make it a hyperbola, and it is not, but a pair of intersecting lines is often referred to as a degenerate hyperbola. Below is a more general case.

Two lines:

$d_1x + e_1y + f_1 = 0$

$d_2x + e_2y + f_2 = 0$

The corresponding second-degree equation:

$(d_1x + e_1y + f_1)(d_2x + e_2y + f_2) = 0$

$d_1d_2x^2 + (d_1e_2 + d_2e_1)xy + e_1e_2y^2 + (d_1f_2 + d_2f_1)x + (e_1f_2 + e_2f_1)y + f_1f_2 = 0$

discriminant = $d_1^2e_2^2 + d_2^2e_1^2 + 2d_1d_2e_1e_2 - 4d_1d_2e_1e_2 = (d_1e_2 - d_2e_1)^2$

This discriminant is positive wherever $d_1e_2 ≠ d_2e_1$, which is to say, in any case involving intersecting lines. The discriminant is zero in the case of parallel lines or coincident lines (a single line). Those cases are sometimes called degenerate parabolas.

Pope
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