Geo3.9

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The distance of a point P(h,k)\, from a pair of lines passing thro'the origin is d units.Show that the equation of the pair of lines is (xk-hy)^{2}=d^{2}(x^{2}+y^{2})\,

Let Q(x_{1},y_{1})\, be any point on the required line.

Therefore area of the triangle OPQ is {\frac  {1}{2}}|x_{1}k-y_{1}h|\,

Triangle OPQ is also equal to {\frac  {1}{2}}d.OQ={\frac  {1}{2}}d{\sqrt  {x_{1}^{{2}}+y_{1}^{{2}}}}\,

Therefore, {\frac  {1}{2}}|x_{1}k-y_{1}h|={\frac  {1}{2}}d{\sqrt  {x_{1}^{{2}}+y_{1}^{{2}}}}\,

Squaring on bothsides,

(x_{1}k-y_{1}h)^{2}=d^{2}(x_{1}^{{2}}+y_{1}^{{2}})\,

The locus of Q is (xk-yh)^{2}=d^{2}(x^{2}+y^{2})\,

This is a homogeneous equation representing a pair of lines passing thro'the origin.


Main Page:Geometry:Straight Lines-II