Question

A rod of length 1 2. m moves with its ends always touching the coordinate axes. The locus of a point P on the rod, which is 0 3. m from the end in contact with x-axis is an ellipse. Find the eccentricity

Answer 1

Length of rod BD = 1.2 m

Let P(x, y) be any point on the Rod such
that PB = 0.3 m

∴ PD = 1.2 – 0.3

= 0.9 m

Let ΔPAB and ΔPCD are similar triangles

In ΔPAB sin θ = `y/(0.3)`

In ΔPCB cos θ = `y/(0.9)`

We know that sin²θ + cos²θ = 1

`y^2/(0.3^2) + x^2/(0.9^2)` = 1

`x^2/(0.9^2) + y^2/(0.3^2)` = 1

a² = 0.9²

b² = 0.3² 

Eccentricity e = `sqrt(1 - "b"^2/"a"^2)`

= `sqrt(1 - 0.3^2/0.9^2)`

= `sqrt(1 - (1/3)^2`

= `sqrt(1 - 1/9)`

= `sqrt(8/9)`

= `(2sqrt(2))/3`