Geo3.7

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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_{2}h_{1}-a_{1}h_{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