Question

The point A is on the positive y-axis and B is on the positive x-axis. P lies on AB such that AP=36 and BP=9. What is the equation of the loci of P?

Answer 1

`y^2 + (x^2)/9 = 81`

Explanation:

Let `A` have coordinates `A(0,alpha)`
Let `B` have coordinates `B(beta,0)`
and `P` has coordinates `P(x,y)`

By Pythagoras for `Delta AOB` we have:

`alpha^2 + beta^2=36^2`

Using similar triangles (or trigonometry) comparing `Delta PRB` with `Delta APQ` we have:

`sin theta = y/9 = (alpha-y)/27`
`:. alpha-y = 3y`
`:. alpha = 4y`

Using similar triangles (or trigonometry) comparing `Delta PRB` with
`Delta APQ` we have:

`cos theta = (beta-x)/9 = x/27`
`:. beta-x = x/3`
`:. beta = (4x)/3`

Substituting `alpha` and `beta` in the first equation gives:

`(4y)^2 + ((4x)/3)^2 = 36^2`
`:. 16y^2 + (16x^2)/9 = 1296`
`:. y^2 + (x^2)/9 = 81`

Which is the required loci