A man running around a race course notes that the sum of the distances of two flagposts from him is always $$10$$ m and the distance between the flag posts is $$8$$ m. Then the area of the path he encloses in square meters is

Question

A man running around a race course notes that the sum of the distances of two flagposts from him is always $$10$$ m and the distance between the flag posts is $$8$$ m. Then the area of the path he encloses in square meters is

Infinity Learn · Accepted Answer

Let $$P(x$$,$$y)$$ be the position of the man at any time.  Let $$S(4$$,$$0)$$ and S′$$($$−$$4$$,$$0)$$ be the fixed flag post, with C as the origin Since $$SP+S$$′$$P=10m$$, i.e., a constant, the locus of P is an  ellipse with S and S′ as foci. Therefore, $$ae=4$$ Also, $$2a=10$$∴ $$e=45$$ Now,     $$b2=a21$$−$$e2$$ or      $$b2=251$$−$$1625=9$$ or      $$b=3$$ Hence, the area of the ellipse is π$$ab=$$π×$$5$$×$$3=15$$π .

$A man running around a race course notes that the sum of the distances of two flagposts from him is always 10 m$
$and the distance between the flag posts is 8 m . Then the area of the path he encloses in square meters is$

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A man running around a race course notes that the sum of the distances of two flagposts from him is always 10 m and the distance between the flag posts is 8 m. Then the area of the path he encloses in square meters is

Remember concepts with our Masterclasses.

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$A man running around a race course notes that the sum of the distances of two flagposts from him is always 10 m$
$and the distance between the flag posts is 8 m . Then the area of the path he encloses in square meters is$