Question

Show that the locus of the mid-point of the distance between the axes of the variable line x cosα + y sinα = p is `1/x^2 + 1/y^2 = 4/p^2` where p is a constant.

Answer 1

Changing the given equation of the line into intercept form

We have `x/(p/(cos alpha)) + y/(p/(sin alpha))` = 1

Which gives the coordinates `p/(cos alpha), 0` and 0, `p/(sin alpha)`

Where the line intersects x-axis and y-axis, respectively.

Let (h, k) denote the mid-point of the line segment joining the points

`p/(cos alpha), 0` and 0, `p/(sin alpha)`

Then h = `p/(2cosalpha)` and k = `p/(2sinalpha)` (Why?)

This gives `cos alpha = p/(2"h")` and `sin alpha = p/(2k)`

Squaring and adding we get

`p^2/(4h^2) + p^2/(4k^2)`

or `1/h^2 + 1/k^2 = 4/p^2`.

Therefore, the required locus is `1/x^2 + 1/y^2 = 4/p^2`.