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Eccentricity of Hyperbola

A bullet is f...

Question

A bullet is fired and it hits a target. An observer in the same plane heard two sounds: the crack of the rifle and the thud of the bullet striking the target at the same instant. Then the locus of the observer is a

A

Ellipse

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B

Hyperbola

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C

Parabola

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D

Circle

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Solution

The correct option is B Hyperbola
Let the target at $B$ and fire point $A$
let the coordinate of the observer is $P$
If velocity of sound is $v_{s}$
Time required to reach the sound of crack to the observer $t_{1}$ and time required to reach the sound of the thud is $t_{2} (t_{1} > t_{2})$
Then $P B = v_{s} t_{2}$
$P A = v_{s} t_{1} \frac{P A}{P B} = \frac{t_{1}}{t_{2}} > 1$
So the path will be hyperbola

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