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Points A and B are 10 km apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is 6 km closer to A than B. Show that the location of the explosion is restricted to a particular curve and find an equation of it.

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#### Solution

As shown in figure, A and B are on both sides of x-axis at Co-ordinates (– 5, 0) and (5, 0)

The distance between A and B is 10. A point C is on the graph at Co-ordinates (x, y)

C is 6 km closer to A than B.

AC = `sqrt((x + 5)^2 + y^2)`

BC = `sqrt((x - 5)^2 + y^2)`

AC – BC = 6

`sqrt((x - 5)^2) + y^2 - sqrt((x + 5)^2 + y^2)` = 6

`sqrt((x - 5)^2 + y^2) = 6 + sqrt((x + 5)^2 + y^2)`

Squaring on both sides we get,

(x – 5)^{2} + y^{2} = `36 + (x + 5)^2 + y^2 + 12(sqrt((x + 5)^2 + y^2))`

x^{2} + 25 – 10x + y^{2} = `x^2 + 10x + y^2 + 36 + 25 + 12sqrt((x + 5)^2 + y^2)`

– 20x – 36 = `12sqrt((x + 5)^2 + y^2)`

(÷ by 4) ⇒ `- 5x - 9 = 3sqrt((x + 5)^2 + y^2)`

Squaring both sides we get,

25x^{2} + 81 + 90x = 9(x^{2} + 25 + 10x + y^{2})

25x^{2} + 81 + 90x – 9x^{2} – 90x – 9y^{2} – 225 = 0

16x^{2} – 9y^{2} – 144 = 0

16x^{2} – 9y^{2} = 144

(÷ by 144) ⇒ `x^2/9 - y^2/16` = 1 is the required equation of hyperbola.

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