Geo3.7

From Exampleproblems

Show that if one of the lines given by $a_1 x^2+2h_1 xy+b_1 y^2=0\,$ coincides with one of the lines of $a_2 x^2+2h_2 xy+b_2 y^2=0\,$ then $(a_1 b_2-a_2 b_1)^2=4(a_2h_1-a_1h_2)(b_1 h_2-b_2 h_1)\,$

Given pairs of lines are $a_1 x^2+2h_1 xy+b_1 y^2=0,a_2 x^2+2h_2 xy+b_2 y^2=0\,$

Let this equations be 1 and 2.

Let $y=mx\,$ be the equation of the common line to 1 and 2.

$y=mx\,$ must satisfy each one of the equations ,hence

$a_1 x^2+2h_1 mx^2+b_1 m^2 x^2=0,b_1 m^2+2h_1 m+a_1=0\,$ equation 3.

$a_2 x^2+2h_2 mx^2+b_2 m^2 x^2=0,b_2 m^2+2h_2 m+a_2=0\,$ equation 4.

Solving 3 and 4,

$\frac{m^2}{2(a_2 h_1-a_1 h_2)}=\frac{m}{a_1 b_2-a_2 b_1}=\frac{1}{2(b_1 h_2-b_2 h_1)}\,$

$m^2=\frac{a_2 h_1-a_1 h_2}{b_1 h_2-b_2 h_1},m=\frac{a_1 b_2-a_2 b_1}{2(b_1 h_2-b_2 h_1)}\,$

Eliminating m,we get

$(a_1 b_2-a_2 b_1)^2=4(a_2 h_1-a_1 h_2)(b_1 h_2-b_2 h_1)\,$

Main Page:Geometry:Straight Lines-II

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