Find the locus of

Given:

Line x sin θ + y cos θ = p

To find:

The locus of the mid-points of the portion of the line x sin θ + y cos θ = p intercepted between the axes.

Explanation:

If the equation of the given line is

x sin θ + y cos θ = p, then the solution is shown below:

The line

x sin θ + y cos θ = p intercepts the axes.

Thus, the coordinate of the poin where the line intercepts x – axis is

Thus, the coordinate of the poin where the line intercepts y – axis is

The midpoint R of the line is given by

R(h, k)

⇒ h , k

Eliminating the sine and cosine terms, we get

⇒

⇒ p²(h² + k²) = 4h²k²

Thus, the locus is given by

p²(x² + y²) = 4x²y²