Question

A stick of length `l` slides with its ends on two mutually perpendicular lines. Find the locus of the middle point of the stick.

Answer 1

Let the stick intercepts x-axis at point `(a,0)` and y-axis at point `(0,b)`
and `(alpha, beta)` are midpoints of the stick. then,
`(a+0)/2 = alpha => a = 2alpha`
`(b+0)/2 = beta=> b= 2beta`
Then, `l^2 = (a-0)^2+(0-b)^2`
Here, `l` is the length of stick.
`=>l^2 = a^2+b^2 `
`=>l^2 = (2alpha)^2+(2beta)^2`
`=>l^2 = 4alpha^2+4beta^2`
`alpha^2+beta^2 = (l/2)^2`
If we replace `(alpha,beta)` by `(x,y)`, then , our equation becomes,
`x^2+y^2 = (l/2)^2`
So, locus will be a circle with center at origin and radius `l/2`.