Fill in the blanks

Locus of the mid-points of the portion of the line x sin θ + y cos θ = p intercepted between the axes is ____.

Given equation of the line is

x sin θ + y cos θ = p …(i)

Let P(h, k) be the midpoint of the given line where it meets the two axis at (a, 0) and (0, b).

Since, (a, 0) lies on eq. (i) then

a sin θ + 0 = p

…(ii)

(0, b) also lies on the eq. (i) then

0 + b cos θ = p

…(iii)

Since, P(h, k) is the midpoint of the given line

⇒ 2h = a

and

⇒ 2k = b

Putting the value of a = 2h in eq. (ii), we get

…(iv)

Putting the value of b = 2k in eq. (ii), we get

…(v)

Squaring and adding eq. (iv) and (v), we get

[∵ sin²θ + cos²θ = 1]

or

or 4x²y² = p²y² + p²x²

or 4x²y² = p²(x² + y²)

Ans. Locus of the mid-points of the portion of the line x sin θ + y cos θ = p intercepted between the axes is 4x²y² = p²(x² + y²)