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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.
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Solution
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`.
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