Question 57 of 70beginner🔧 ApplyShort Answer2 marks

In each of the Exercises 7 to 15, find the equations of the hyperbola satisfying the given conditions: Foci (± 5, 0), the transverse axis is of length 8.

Correct Answer

The equation of the hyperbola is x216−y29=1\frac{x^2}{16} - \frac{y^2}{9} = 1.

Exercise: EXERCISE 10.4 | Q: 10 | (Page 27)
For More Understanding

Explanation

Since the foci are at (±5, 0), they lie on the x-axis, so the hyperbola has its transverse axis along the x-axis. The standard form of such a hyperbola is x²/a² - y²/b² = 1. From the foci coordinates, c = 5. From the transverse axis length of 8, we get 2a = 8, so a = 4. Using the relation c² = a² + b² for hyperbolas, we find b² = c² - a² = 25 - 16 = 9.

Solution Steps

  1. Step 1: Identify that foci (±5, 0) lie on x-axis, so hyperbola equation is of form x²/a² - y²/b² = 1

  2. Step 2: From foci, c = 5

  3. Step 3: From transverse axis length = 8, we get 2a = 8, so a = 4

  4. Step 4: Using relation c² = a² + b², calculate b² = 25 - 16 = 9

  5. Step 5: Write equation: x²/16 - y²/9 = 1

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