Question 39 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 (± 6, 0), foci (± 4, 0)

Correct Answer

The equation of the ellipse is x236+y220=1\frac{x^2}{36} + \frac{y^2}{20} = 1.

Exercise: EXERCISE 10.3 | Q: 12 | (Page 20)
For More Understanding

Explanation

This question is from Exercise 10.3, which deals with finding equations of ellipses. The student needs to identify that since vertices and foci are both on the x-axis, the major axis is horizontal. Using the standard form x²/a² + y²/b² = 1, they calculate a from vertices and c from foci, then use the relationship c² = a² - b² to find b².

Solution Steps

  1. Step 1: Since vertices (± 6, 0) and foci (± 4, 0) are on the x-axis, the equation is of the form x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, where a is the semi-major axis.

  2. Step 2: From vertices (± 6, 0), we get a = 6, so a² = 36.

  3. Step 3: From foci (± 4, 0), we get c = 4, so c² = 16.

  4. Step 4: Using the relation c² = a² - b², we get: 16=36−b216 = 36 - b^2, therefore b² = 20.

  5. Step 5: Substituting a² = 36 and b² = 20 in the standard equation: x236+y220=1\frac{x^2}{36} + \frac{y^2}{20} = 1.

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