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0H UTCHIpse Whose foci is ($3,0) and which pass through the point (4,1)On a level plain, the crack of the rifle and the thud of the ball striking the targetare heard at the same instant. Prove that the locus of the bearer is a hyperbola.Prove that the equation of the nornhol. 1 |
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Answer» Suppose A to be the target and B to be the firing point and let the hearer be at a point P. If V1and V2be the velocity of the sound and bullet respectively, then the time taken by the bullet in reaching from B to the target A = BA/v2and the time taken by the sound in reaching from A to P = AP/v1. Again the time taken by the sound in reaching from B to P = BP/v1. As by hypothesis the sound reaches the hearer at P, simultaneously, the time taken by the sound to reach upto the position P from then target together with the time taken by the bullet to reach the target must be same as the time taken by the sound in reaching from the rifle upto P. Hence BA/v2+ AP/v1= BP/v1 ⇒BA/v2= BP/v1– AP/v1= 1/v1(BP – AP) ⇒BP – AP = v1/v2AB As v1, v2and AB are constants, hence BP –AP = Constant Therefore the locus of the point P is the hyperbola having foci at A and B and transverse axis equal to v1/v2AB |
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