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

Algebra ->  Length-and-distance -> SOLUTION: The distance of a point P(h,k) from a pair of straight lines passing through origin is 'd' units. Show that the equation of the pair of lines is (xk - hy)^2=d^2(x^2+y^2).      Log On


   

Question 1046270: The distance of a point P(h,k) from a pair of straight lines passing through origin is 'd' units. Show that the equation of the pair of lines is (xk - hy)^2=d^2(x^2+y^2).
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Let the origin be point O.
Let the points R and S be where the line M perpendicular to line OP and passing through P intersect the two lines. (Note that line OP actually bisects the angle between the pair of lines.)
Let the point S be (x,y), and let G be such that the segment PG is perpendicular to line OS.
Then by similarity of triangles,
abs%28PS%29%2Fabs%28PG%29+=+abs%28OS%29%2Fabs%28OP%29+
===>
===>
===> ++abs%28hy-kx%29+%2Fd+=+sqrt%28x%5E2%2By%5E2%29
===> ++abs%28hy-kx%29+=d%2Asqrt%28x%5E2%2By%5E2%29
<===> ++abs%28hy-kx%29%5E2+=d%5E2%2A%28x%5E2%2By%5E2%29, or equivalently,
++%28hy-kx%29%5E2+=d%5E2%2A%28x%5E2%2By%5E2%29
and this finishes the solution.