Question

ABCD is a variable rectangle having its sides parallel to fixed directions. The vertices B and D lie on x = a and x = a and A lies on the line y = 0. Find the locus of point C.

Answer 1

See Fig. Let A be (x₁, 0), B be (a, y₂) and D be (−a, y₃). We are given that AB and AD have fixed directions and hence their slopes are constant, say, m₁ and m₂. Therefore,

y₂/a - x₁ = m₁ and y₃/-a - x₁ = m₂

Further, m₁m₂ = −1 since ABCD is rectangle.

y₂/a - x₁ = m₁ and y₃/a + x₁ = 1/m₁  .....(1)

Let the coordinates of C be (α ,β). Now, Midpoint of BD ≡ Midpoint of AC

This implies that

By Eqs. (1) and (2), we have

-(m₁² - 1)α + m₁β =(m₁² + 1)a

Therefore, the locus of point C is

m₁y = (m²₁ + 1)a + (m²₁ - 1)x