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Question
Find the equation of the set of all points the sum of whose distances from the points (3, 0) and (9, 0) is 12.
Solution
Let (x, y) be any point.
Given points are (3, 0) and (9, 0)
We have `sqrt((x - 3)^2 + (y - 0)^2) + sqrt((x - 9)^2 + (y - 0)^2)` = 12
⇒ `sqrt(x^2 + 9 - 6x + y^2) + sqrt(x^2 + 81 - 18x + y^2)` = 12
Putting x2 + 9 – 6x + y2 = k
⇒ `sqrt(k) + sqrt(72 - 12x + k)` = 12
⇒ `sqrt(72 - 12x + k) = 12 - sqrt(k)`
Squaring both sides, we have
⇒ 72 – 12x + k = `144 + k - 24sqrt(k)`
⇒ `24sqrt(k)` = 144 – 72 + 12x
⇒ `24sqrt(k)` = 72 + 12x
⇒ `2sqrt(k)` = 6 + x
Again squaring both sides, we get
4k = 36 + x2 + 12x
Putting the value of k, we have
4(x2 + 9 – 6x + y2) = 36 + x2 + 12x
⇒ 4x2 + 36 – 24x + 4y2 = 36 + x2 + 12x
⇒ 3x2 + 4y2 – 36x = 0
Hence, the required equation is 3x2 + 4y2 – 36x = 0.
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