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A man running a racecourse notes that th...

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.

A

`15pi`

B

`20pi`

C

`27pi`

D

`30pi`

Text Solution

Verified by Experts

Let P(x,y) be the postion of the man at any time.
Let S(4,0) and S' (-4,0) be the fixed flag post, with C as the orign.
Sicne S+P+SP=10m i.e., a constant , the locus of P is an ellipse with S and S as foci. Therefore,
ae=4
Alos, 2a=10
`:. e=(4)/(5)`
Now, `b^(2)=a^(2)(1-e^(2))`
or `b^(2)=25(1-(16)/(25))=9`
or b=3
Hence, the area of the ellipse is `piab=pixx5xx3=15pi`
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