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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
Solution
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
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