First slide

Ellipse

Question

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

Moderate

A

$15 π$
B

$20 π$
C

$27 π$
D

$30 π$

Solution

$Let P (x, y) be the position of the man at any time.$
$Let S (4, 0) and S^{'} (- 4, 0) be the fixed flag post, with C as the origin$
$Since S P + S^{'} P = 10 m, i.e., a constant, the locus of P is an$
$ellipse with S and S^{'} as foci. Therefore,$
$\begin{aligned} a e = 4 \\ Also, 2 a = 10 \\ ∴ e = \frac{4}{5} \end{aligned}$
$\begin{array}{ll} Now, b^{2} = a^{2} (1 - e^{2}) \\ or b^{2} = 25 (1 - \frac{16}{25}) = 9 \\ or b = 3 \end{array}$
$Hence, the area of the ellipse is π a b = π \times 5 \times 3 = 15 π .$

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