Q. 185.0( 3 Votes )

# 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

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