Question 36 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 | (Page 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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