Question 29 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: x²/4 + y²/25 = 1

Correct Answer

For the ellipse x²/4 + y²/25 = 1:

Foci: (0, 21\sqrt{21}) and (0, -21\sqrt{21})

Vertices: (0, 5) and (0, -5)

Length of major axis: 10 units

Length of minor axis: 4 units

Eccentricity: 21\sqrt{21}/5

Length of latus rectum: 8/5 units

Exercise: EXERCISE 10.3 | Q: 2 | (Page 20)
For More Understanding

Explanation

This question is from Exercise 10.3, Question 2. The solution follows the standard method shown in Example 9 and the worked example in the context. Since the denominator of y²/25 (which is 25) is larger than the denominator of x²/4 (which is 4), the major axis is along the y-axis. The student compares with the standard form x²/b² + y²/a² = 1 to find a = 5 and b = 2.

Solution Steps

  1. Step 1: Compare denominators: 25 > 4, so major axis is along y-axis.

  2. Step 2: Compare with standard form x²/b² + y²/a² = 1, we get b² = 4, a² = 25, so b = 2, a = 5.

  3. Step 3: Find c = a2−b2\sqrt{a² - b²} = 25−4\sqrt{25 - 4} = 21\sqrt{21}

  4. Step 4: Find eccentricity e = c/a = 21\sqrt{21}/5

  5. Step 5: Foci are (0, ±c) = (0, ±21\sqrt{21}); Vertices are (0, ±a) = (0, ±5)

  6. Step 6: Length of major axis = 2a = 10 units; Length of minor axis = 2b = 4 units

  7. Step 7: Length of latus rectum = 2b²/a = 2(4)/5 = 8/5 units

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