Question

The sum of the distance of a moving point from the points (4, 0) and (−4, 0) is always 10 units. Find the equation of the locus of the moving point

Answer 1

Let A be (4, 0) and B be (−4, 0).

Let the moving point be P(h, k)

Given PA + PB = 10

`sqrt(("h" - 4)^2 + ("k" - 0)^2) + sqrt(("h" + 4)^2 + ("k" -  0)^2` = 10

`sqrt(("h" - 4)^2 + "k"^2) + sqrt(("h" + 4)^2 + "k"^2)` = 10

`sqrt(("h" - 4)^2 + "k") = 10 - sqrt(("h" + 4)^2 + "k"^2)`

Squaring on both sides

(h − 4)² + k² = `[10 - sqrt(("h" + 4)^2 + "k"^2)]^2` 

h² − 8h + 16 + k² = `100 - 20sqrt(("h" + 4)^2 + "k"^2) + ("h" + 4)^2 + "k"^2`

h² − 8h + 16 + k² = `100 - 20 sqrt(("h" + 4)^2 + "k"^2) + "h"^2 + 8"h" + 16 + "k"^2`

− 16h −100 = `- 20 sqrt(("h" + 4)^2 + "k"^2)`

4h + 25 = `5 sqrt(("h" + 4)^2 + "k"^2)`

(4h + 25)² = `25[("h" + 4)^2 + "k"^2]`

16h² + 200h + 625 = 25[h² + 8h + 16 + k²]

16h² + 200h + 625 = 25h² + 200h + 400 + 25k²

9h² + 26k² = 225

`(9"h"^2)/225 + (25"k"^2)/225` = 1

`("h"^2)/25 + "k"^2/9` = 1

∴ The locus of P(h, k) is  `x^2/25 + y^2/9` = 1