Question 54 of 70intermediate🔧 ApplyShort Answer2 marks

In each of the Exercises 1 to 6, find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas: 49y² - 16x² = 784

Correct Answer

The given equation of the hyperbola is 49y² - 16x² = 784. Dividing both sides by 784: y²/16 - x²/49 = 1

This is of the form y²/a² - x²/b² = 1, where a² = 16 and b² = 49. So, a = 4 and b = 7.

Since the positive term is y², the transverse axis is along y-axis.

Vertices: (0, ±a) = (0, ±4)

Foci: c² = a² + b² = 16 + 49 = 65, so c = 65\sqrt{65} Foci are at (0, ±65\sqrt{65})

Eccentricity: e = c/a = 65\sqrt{65}/4

Length of latus rectum: 2b²/a = 2(49)/4 = 49/2

Exercise: EXERCISE 10.4 | Q: 6 | (Chapter: 27)
For More Understanding

Explanation

This question from Exercise 10.4 requires converting the hyperbola equation to standard form. The key insight is recognizing that when y² has the positive coefficient, the transverse axis lies along the y-axis. The context from Example 14 and Example 15 demonstrates the standard procedure for finding vertices, foci, eccentricity, and latus rectum for hyperbolas of the form y²/a² - x²/b² = 1.

Solution Steps

  1. Step 1: Divide the equation 49y² - 16x² = 784 by 784 to get y²/16 - x²/49 = 1

  2. Step 2: Identify a² = 16, b² = 49, so a = 4, b = 7

  3. Step 3: Find c using c² = a² + b² = 65, so c = 65\sqrt{65}

  4. Step 4: Vertices are (0, ±a) = (0, ±4)

  5. Step 5: Foci are (0, ±c) = (0, ±65\sqrt{65})

  6. Step 6: Eccentricity e = c/a = 65\sqrt{65}/4

  7. Step 7: Length of latus rectum = 2b²/a = 49/2

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