Question 42 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: Length of major axis 26, foci (± 5, 0)

Correct Answer

The equation of the ellipse is x2169+y2144=1\frac{x^2}{169} + \frac{y^2}{144} = 1.

Exercise: EXERCISE 10.3 | Q: 15 | (Page 20)
For More Understanding

Explanation

This question is similar to Example 11 from the textbook. Since the foci are at (±5, 0), they lie on the x-axis, which means the major axis is along the x-axis. The equation of the ellipse will be of the form x²/a² + y²/b² = 1. The length of major axis is 26, so 2a = 26 giving a = 13. The foci are at (±5, 0), so c = 5. Using the relation c² = a² - b², we substitute the values to find b², and then write the final equation.

Solution Steps

  1. Step 1: Since foci (±5, 0) are on the x-axis, the major axis is along the x-axis. The equation is of the form x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.

  2. Step 2: Given length of major axis = 26, so 2a = 26, which gives a = 13 and a² = 169.

  3. Step 3: Foci are at (±5, 0), so c = 5 and c² = 25.

  4. Step 4: Using the relation c² = a² - b², we get 25=169−b225 = 169 - b^2, which gives b² = 144.

  5. Step 5: Substituting a² = 169 and b² = 144 in the standard form, the equation is x2169+y2144=1\frac{x^2}{169} + \frac{y^2}{144} = 1.

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