Understanding the Racecourse Path: An Ellipse

The question describes a man running around a racecourse. He observes that the sum of the distances from his position to two fixed flag-posts is always constant. This is the defining property of an ellipse. The two flag-posts are located at the foci (plural of focus) of the ellipse.

Let the two flag-posts be $F_1$ and $F_2$. Let the man's position at any point be $P$. The problem states that the sum of the distances from $P$ to $F_1$ and $F_2$ is constant:

$\text{Distance}(P, F_1) + \text{Distance}(P, F_2) = \text{constant}$

This constant sum is equal to the length of the major axis of the ellipse, which is denoted by $2a$.

Given in the problem:

• Sum of the distance of two flag-posts from him is always 10 m. So, $2a = 10$ m.
• The distance between the flag-posts (the foci) is 8 m. The distance between the two foci of an ellipse is denoted by $2c$. So, $2c = 8$ m.

Calculating Ellipse Parameters: Semi-major and Semi-minor Axes

From the given information, we can find the values of the semi-major axis ($a$) and the distance from the center to the focus ($c$).

• $2a = 10$ m $\implies a = \frac{10}{2} = 5$ m
• $2c = 8$ m $\implies c = \frac{8}{2} = 4$ m

For an ellipse, there is a relationship between the semi-major axis ($a$), the semi-minor axis ($b$), and the distance from the center to the focus ($c$). This relationship is given by the equation:

$\hspace{2em} a^2 = b^2 + c^2$

We need to find the semi-minor axis ($b$) to calculate the area of the ellipse. We can rearrange the formula to solve for $b^2$:

$\hspace{2em} b^2 = a^2 - c^2$

Now, substitute the values of $a$ and $c$ that we found:

$\hspace{2em} b^2 = (5)^2 - (4)^2$
$\hspace{2em} b^2 = 25 - 16$
$\hspace{2em} b^2 = 9$

Taking the square root of both sides to find $b$:

$\hspace{2em} b = \sqrt{9}$
$\hspace{2em} b = 3$ m

So, the semi-minor axis of the ellipse is 3 metres.

Calculating the Area of the Enclosed Path

The path the man encloses is the area of the ellipse. The formula for the area of an ellipse is given by:

$\hspace{2em} \text{Area} (A) = \pi \times a \times b$

We have found the values for the semi-major axis ($a = 5$ m) and the semi-minor axis ($b = 3$ m). Substitute these values into the area formula:

$\hspace{2em} A = \pi \times 5 \times 3$
$\hspace{2em} A = 15\pi$ square metres

Thus, the area of the path the man encloses is $15\pi$ square metres.

Comparing this result with the given options, we find that $15\pi$ square metres is one of the choices.

Revision Table: Ellipse Properties

Additional Information: Conic Sections and Ellipses

Ellipses are one type of conic section, which are curves formed by the intersection of a plane and a double cone. Other conic sections include circles, parabolas, and hyperbolas.

Key features of an ellipse:

• Eccentricity ($e$): A measure of how elongated the ellipse is. It is defined as $e = c/a$. For an ellipse, $0 \le e < 1$. A circle is a special case of an ellipse where $e=0$ (which means $c=0$, so the foci coincide with the center).
• Directrices: For every point on the ellipse, the ratio of its distance to a focus and its distance to a fixed line (called the directrix) is constant and equal to the eccentricity $e$.

The property used in this problem, where the sum of distances to the foci is constant, is a fundamental definition of an ellipse and is often used in practical applications, such as the design of "whispering galleries" where sound waves reflect from the walls to the other focus.