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Question
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.
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 + y2 = `36 + (x + 5)^2 + y^2 + 12(sqrt((x + 5)^2 + y^2))`
x2 + 25 – 10x + y2 = `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,
25x2 + 81 + 90x = 9(x2 + 25 + 10x + y2)
25x2 + 81 + 90x – 9x2 – 90x – 9y2 – 225 = 0
16x2 – 9y2 – 144 = 0
16x2 – 9y2 = 144
(÷ by 144) ⇒ `x^2/9 - y^2/16` = 1 is the required equation of hyperbola.
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