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