Question 50 of 70beginner🔧 ApplyShort Answer2 marks

In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse: x²/49 + y²/36 = 1

Correct Answer

Given equation of ellipse is x²/49 + y²/36 = 1.

Step 1: Since denominator of x²/49 (which is 49) is larger than denominator of y²/36 (which is 36), the major axis is along the x-axis. Comparing with standard form x²/a² + y²/b² = 1, we get a = 7 and b = 6.

Step 2: Find c using c = a2−b2\sqrt{a² - b²} = 49−36\sqrt{49 - 36} = 13\sqrt{13}

Coordinates of foci: (±13\sqrt{13}, 0) i.e., (13\sqrt{13}, 0) and (-13\sqrt{13}, 0)

Vertices: (±a, 0) i.e., (7, 0) and (-7, 0)

Length of major axis: 2a = 2(7) = 14 units

Length of minor axis: 2b = 2(6) = 12 units

Eccentricity: e = c/a = 13\sqrt{13}/7

Length of latus rectum: 2b²/a = 2(36)/7 = 72/7 units

Exercise: EXERCISE 10.3 | Q: 5 | (Chapter: 20)
For More Understanding

Explanation

This question follows the same pattern as Example 9 in the textbook. The key is to identify which denominator is larger to determine the orientation of the major axis. Since 49 > 36, the major axis is along x-axis. The formula c = a2−b2\sqrt{a² - b²} gives the distance from center to each focus. All other values follow from standard formulas for ellipses.

Solution Steps

  1. Step 1: Compare denominators - 49 > 36, so major axis along x-axis

  2. Step 2: Identify a² = 49, b² = 36, therefore a = 7, b = 6

  3. Step 3: Calculate c = a2−b2\sqrt{a² - b²} = 49−36\sqrt{49 - 36} = 13\sqrt{13}

  4. Step 4: Foci at (±c, 0) = (±13\sqrt{13}, 0)

  5. Step 5: Vertices at (±a, 0) = (±7, 0)

  6. Step 6: Major axis = 2a = 14 units, Minor axis = 2b = 12 units

  7. Step 7: Eccentricity e = c/a = 13\sqrt{13}/7

  8. Step 8: Latus rectum = 2b²/a = 72/7 units

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