Question 59 of 70intermediate🔧 ApplyShort Answer2 marks

In each of the Exercises 7 to 15, find the equations of the hyperbola satisfying the given conditions: Foci (± 35\sqrt{5}, 0), the latus rectum is of length 8.

Correct Answer

x²/25 - y²/20 = 1

Exercise: EXERCISE 10.4 | Q: 12 | (Page 27)
For More Understanding

Explanation

This question is from Exercise 10.4, Question 12. The student needs to find the hyperbola equation given foci and latus rectum length. Since foci are on the x-axis at (±35\sqrt{5}, 0), the hyperbola is of the standard form x²/a² - y²/b² = 1. Using the relationship c² = a² + b² and latus rectum formula 2b²/a, we can solve for a and b.

Solution Steps

  1. Step 1: Since foci are at (±35\sqrt{5}, 0), the hyperbola is of the form x²/a² - y²/b² = 1 with c = 35\sqrt{5}.

  2. Step 2: Given c = 35\sqrt{5}, we have c² = 45.

  3. Step 3: Latus rectum = 2b²/a = 8, so b² = 4a.

  4. Step 4: Using c² = a² + b², we get 45 = a² + 4a.

  5. Step 5: Solving: a² + 4a - 45 = 0, which gives (a + 9)(a - 5) = 0.

  6. Step 6: Since a > 0, we take a = 5. Then b² = 4(5) = 20.

  7. Step 7: Therefore, the equation is x²/25 - y²/20 = 1.

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