Question 67 of 70intermediate🔧 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: 4x² + 9y² = 36

Correct Answer

The given equation of the ellipse is 4x² + 9y² = 36.

Step 1: Converting to standard form by dividing both sides by 36: x²/9 + y²/4 = 1

Step 2: Comparing with the standard equation x²/a² + y²/b² = 1: a² = 9, so a = 3 b² = 4, so b = 2

Since a > b, the major axis is along the x-axis.

Step 3: Finding c: c = a2−b2\sqrt{a² - b²} = 9−4\sqrt{9 - 4} = 5\sqrt{5}

Results:

  • Coordinates of foci: (±5\sqrt{5}, 0) i.e., (5\sqrt{5}, 0) and (-5\sqrt{5}, 0)
  • Vertices: (±3, 0) i.e., (3, 0) and (-3, 0)
  • Length of major axis: 2a = 6 units
  • Length of minor axis: 2b = 4 units
  • Eccentricity: e = c/a = 5\sqrt{5}/3
  • Length of latus rectum: 2b²/a = 8/3 units
Exercise: EXERCISE 10.3 | Q: 9 | (Chapter: 20)
For More Understanding

Explanation

This question requires converting the ellipse equation to standard form. Following the method shown in Example 10 of the textbook, we divide by 36 to get x²/9 + y²/4 = 1. Since the denominator under x² (which is 9) is larger than the denominator under y² (which is 4), the major axis lies along the x-axis. Using the standard formulas: c = a2−b2\sqrt{a² - b²}, e = c/a, and latus rectum = 2b²/a, we compute all required values.

Solution Steps

  1. Step 1: Divide 4x² + 9y² = 36 by 36 to get x²/9 + y²/4 = 1

  2. Step 2: Identify a² = 9, b² = 4, so a = 3, b = 2

  3. Step 3: Calculate c = a2−b2\sqrt{a² - b²} = 9−4\sqrt{9-4} = 5\sqrt{5}

  4. Step 4: Find foci (±c, 0) = (±5\sqrt{5}, 0)

  5. Step 5: Find vertices (±a, 0) = (±3, 0)

  6. Step 6: Calculate eccentricity e = c/a = 5\sqrt{5}/3

  7. Step 7: Calculate latus rectum = 2b²/a = 8/3

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