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

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