Geo3.26

From Example Problems
Jump to: navigation, search

Show that the lines joining the origin to the points of intersection of two curves ax^{2}+2hxy+by^{2}+2gx=0,a_{1}x^{2}+2h_{1}xy+by_{1}^{{2}}+2g_{1}x=0\, will be at right angles to one another if g(a_{1}+b_{1})=g_{1}(a+b)\,

Let the curves be 1 and 2 cut in A and B.

From 2,2x=(a_{1}x^{2}+2h_{1}xy+b_{1}y^{2})+(-g_{1})\,

Equation to OA and OB is obtained by homogenising 1 and 2.

Therefore, equation of OA and OB is ax^{2}+2hxy+by^{2}+g[{\frac  {a_{1}x^{2}+2h_{1}xy+b_{1}y^{2}}{-g_{1}}}]=0\,

Since \angle AOB=90^{\circ }\,

Coefficient of x^{2}\, + coefficient of y^{2}\, is zero.

[a+{\frac  {ga_{1}}{-g_{1}}}]+[b+{\frac  {gb_{1}}{-g_{1}}}]=0\,

-g_{1}a+ga_{1}-bg_{1}+gb_{1}=0\,

g(a_{1}+b_{1})=g_{1}(a+b)\,


Main Page:Geometry:Straight Lines-II