Tamil Nadu Board of Secondary EducationHSC Arts Class 12

# 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 - Mathematics

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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.

#### 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.

Concept: Real Life Applications of Conics
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Chapter 5: Two Dimensional Analytical Geometry-II - Exercise 5.5 [Page 215]

#### APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 12th Mathematics Volume 1 and 2 Answers Guide
Chapter 5 Two Dimensional Analytical Geometry-II
Exercise 5.5 | Q 10 | Page 215
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