Question

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

Answer 1

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).

Explanation:

Given equation of the line is

x cos θ + y sin θ = p   ......(i)

Let C(h, k) be the mid-point of the given line AB where it meets the two-axis at A(a, 0) and B(0, b).

Since (a, 0) lies on equation (i) then

a cos θ + 0 = p

⇒ `a = p/costheta`  ......(ii)

B(0, b) also lies on the equation (i) then

0 + b sin θ = p

⇒ `b = p/sintheta`  ......(iii)

Since C(h, k) is the mid-point of AB

∴ `h = (0 + a)/2`

⇒ a = 2h

And k = `(b + 0)/2`

⇒ b = 2k

Putting the values of a and b is equation (ii) and (iii) we get

2h = `p/costheta`

⇒ cos θ = `p/(2h)`   ......(iv)

And 2h = `p/sintheta`

⇒ sin θ = `p/(2k)`   ......(v)

Squaring and adding equation (iv) and (v) we get

⇒ cos²θ + sin²θ = `p^2/(4h^2) + p^2/(4k^2)`

⇒ 1 = `p^2/(4h^2) + p^2/(4k^2)`

So, the locus of the mid-point is 

1 = `p^2/(4x^2) + p^2/(4y^2)`

⇒ 4x²y² = p²(x² + y²)