Question 31 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: x²/25 + y²/100 = 1

Correct Answer

The given equation is x²/25 + y²/100 = 1. Since the denominator of y²/100 is larger than the denominator of x²/25, the major axis is along the y-axis. Comparing with the standard equation x²/b² + y²/a² = 1, we have a² = 100 and b² = 25, so a = 10 and b = 5.

c = a2−b2\sqrt{a² - b²} = 100−25\sqrt{100 - 25} = 75\sqrt{75} = 53\sqrt{3} and e = c/a = 53\sqrt{3}/10 = 3\sqrt{3}/2.

Coordinates of foci: (0, ±53\sqrt{3}) Vertices: (0, ±10) Length of major axis: 2a = 20 units Length of minor axis: 2b = 10 units Eccentricity: 3\sqrt{3}/2 Length of latus rectum: 2b²/a = 2(25)/10 = 5 units

Exercise: EXERCISE 10.3 | Q: 4 | (Page 20)
For More Understanding

Explanation

This question tests understanding of ellipse properties when the major axis is vertical. The key insight is recognizing that when the denominator under y² is larger, the major axis lies along the y-axis, requiring use of the form x²/b² + y²/a² = 1. Students must correctly identify a and b, then apply standard formulas for foci, vertices, axes, eccentricity, and latus rectum.

Solution Steps

  1. Step 1: Identify that denominator 100 > 25, so major axis is along y-axis

  2. Step 2: Compare with x²/b² + y²/a² = 1 to get a = 10, b = 5

  3. Step 3: Calculate c = a2−b2\sqrt{a² - b²} = 75\sqrt{75} = 53\sqrt{3}

  4. Step 4: Find foci (0, ±c) = (0, ±53\sqrt{3}) and vertices (0, ±a) = (0, ±10)

  5. Step 5: Calculate eccentricity e = c/a = 3\sqrt{3}/2 and latus rectum = 2b²/a = 5

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