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प्रश्न
Find the equation of the set of all points the sum of whose distances from the points (3, 0) and (9, 0) is 12.
उत्तर
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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