Condition for Pairs of Lines through Origin to be Harmonic Conjugates/Homogeneous Quadratic Equation Form

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Theorem

Consider the two homogeneous quadratic equations:

\(\text {(E1)}: \quad\) \(\ds a_1 x^2 + 2 h_1 x y + b_1 y^2\) \(=\) \(\ds 0\)
\(\text {(E2)}: \quad\) \(\ds a_2 x^2 + 2 h_2 x y + b_2 y^2\) \(=\) \(\ds 0\)

each representing two straight lines through the origin.


Then the two straight lines represented by $(\text E1)$ are harmonic conjugates of the two straight lines represented by $(\text E2)$ if and only if:

$a_1 b_2 + a_2 b_1 - 2 h_1 h_2 = 0$


Proof

From Homogeneous Quadratic Equation represents Two Straight Lines through Origin, $(\text E1)$ and $(\text E2)$ represent straight lines through the origin if and only if:

\(\ds h_1^2 - a_1 b_1\) \(>\) \(\ds 0\)
\(\ds h_2^2 - a_2 b_2\) \(>\) \(\ds 0\)


Let the two straight lines represented by $(\text E1)$ be defined by the equations:

\(\ds y\) \(=\) \(\ds \lambda_1 x\)
\(\ds y\) \(=\) \(\ds \mu_1 x\)


Let the two straight lines represented by $(\text E2)$ be defined by the equations:

\(\ds y\) \(=\) \(\ds \lambda_2 x\)
\(\ds y\) \(=\) \(\ds \mu_2 x\)


Then we can write the Condition for Pairs of Lines through Origin to be Harmonic Conjugates as:

$(1): \quad 2 \paren {\lambda_1 \mu_1 + \lambda_2 \mu_2} = \paren {\lambda_1 + \mu_1} \paren {\lambda_2 + \mu_2}$


We can express $a_1 x^2 + 2 h_1 x y + b_1 y^2 = 0$ as:

$b_1 \paren {y - \lambda_1 x} \paren {y - \mu_1 x} = 0$

and $a_2 x^2 + 2 h_2 x y + b_2 y^2 = 0$ as:

$b_2 \paren {y - \lambda_2 x} \paren {y - \mu_2 x} = 0$


from which we obtain via Sum of Roots of Quadratic Equation and Product of Roots of Quadratic Equation:

\(\ds \lambda_1 + \mu_1\) \(=\) \(\ds -\dfrac {2 h_1} {b_1}\)
\(\ds \lambda_2 + \mu_2\) \(=\) \(\ds -\dfrac {2 h_2} {b_2}\)
\(\ds \lambda_1 \mu_1\) \(=\) \(\ds \dfrac {a_1} {b_1}\)
\(\ds \lambda_2 \mu_2\) \(=\) \(\ds \dfrac {a_2} {b_2}\)


Therefore $(1)$ reduces to:

\(\ds 2 \paren {\dfrac {a_1} {b_1} + \dfrac {a_2} {b_2} }\) \(=\) \(\ds \dfrac {4 h_1 h_2} {b_1 b_2}\)
\(\ds \leadsto \ \ \) \(\ds a_1 b_2 + a_2 b_1 - 2 h_1 h_2\) \(=\) \(\ds 0\)

$\blacksquare$


Sources

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