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 and B. Show that the location of the explosion is restricted to a particular curve and find an equation of it.
Answers
Answer:
X²/9 - y²/16 = 1
Step-by-step explanation:
As shown in figure, A & B are on both sides of X Axis at coordinates (-5,0) and (5, 0).
The distance between A & B is 10.
A point C is on the graph at coordinates (x,y).
C is 6 km (or units) closer to A than B.
Distance between C & A = √((x + 5)² + (y²))
Distance between C & B = √((x - 5)² + (y²))
From the question, distance between C & B is longer by 6.
Hence
√((x - 5)² + (y²)) - √((x + 5)² + (y²)) = 6
√((x - 5)² + (y²)) = 6 + √((x + 5)² + (y²))
Squaring both sides ,we get
((x - 5)² + (y²)) = 36 + ((x + 5)² + (y²)) + 12√((x + 5)² + (y²))
X² + 25 – 10x + y² = x² + 10x + y² + 61 + 12√((x + 5)² + (y²))
Simplifying we get
-5x – 9 = 3√((x + 5)² + (y²))
Squaring both sides, we get
25x² + 81 + 90x = 9x² + 225x + 90 + 9y²
16x² – 9y² = 144
X²/9 - y²/16 = 1
This is a equation of a hyperbola.
