Question 38 of 70beginner🔧 ApplyShort Answer2 marks

In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions: Vertices (0, ± 13), foci (0, ± 5)

Correct Answer

The equation of the ellipse is x2144+y2169=1\frac{x^2}{144} + \frac{y^2}{169} = 1.

Exercise: EXERCISE 10.3 | Q: 11 | (Page 20)
For More Understanding

Explanation

This question tests understanding of ellipse equations when the major axis is along the y-axis. The key insight is recognizing that when vertices and foci have coordinates (0, ±a) and (0, ±c) respectively, the major axis lies on the y-axis, and the equation takes the form x²/b² + y²/a² = 1. The relationship c² = a² - b² is then used to find b.

Solution Steps

  1. Step 1: Since vertices (0, ±13) and foci (0, ±5) lie on the y-axis, the major axis is along the y-axis. The equation is of the form x2b2+y2a2=1\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1.

  2. Step 2: From the vertices (0, ±13), we get a = 13 (semi-major axis). From the foci (0, ±5), we get c = 5.

  3. Step 3: Using the relation c2=a2−b2c^2 = a^2 - b^2, we get 25=169−b225 = 169 - b^2, which gives b² = 144.

  4. Step 4: Substituting a² = 169 and b² = 144, the equation of the ellipse is x2144+y2169=1\frac{x^2}{144} + \frac{y^2}{169} = 1.

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