Question 69 of 70intermediate🔧 ApplyLong Answer4 marks

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.

Correct Answer

The equation of the path traced by the man is

x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1

or

9x2+25y2=2259x^2 + 25y^2 = 225
Exercise: Miscellaneous Exercise on Chapter 10 | Q: 7 | (Page 29)
For More Understanding

Explanation

This question is based on the definition of an ellipse. According to the NCERT textbook, an ellipse is the set of all points in a plane, the sum of whose distances from two fixed points (called foci) is constant. Here, the two flag posts act as the foci of the ellipse, and the man traces an elliptical path where the sum of distances from both flag posts remains constant at 10 m.

Solution Steps

  1. Step 1: Identify that the path is an ellipse since the sum of distances from two fixed points (flag posts) is constant at 10 m.

  2. Step 2: Let the flag posts be the foci of the ellipse. Place the midpoint between them at the origin with foci at (−c,0)(-c, 0) and (c,0)(c, 0).

  3. Step 3: Given distance between flag posts = 8 m, so 2c=82c = 8, therefore c = 4.

  4. Step 4: The sum of distances from any point on ellipse to both foci = 2a=102a = 10 m, therefore a = 5 and a² = 25.

  5. Step 5: Using the ellipse relation c2=a2−b2c^2 = a^2 - b^2, we get 16=25−b216 = 25 - b^2, therefore b² = 9.

  6. Step 6: The standard equation of ellipse with foci on x-axis is

    x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

  7. Step 7: Substituting a2=25a^2 = 25 and b2=9b^2 = 9, the equation is

    x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1

    or

    9x2+25y2=2259x^2 + 25y^2 = 225
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