Question 51 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: 16x² - 9y² = 576

Correct Answer

Foci: (±10, 0); Vertices: (±6, 0); Eccentricity: 5/3; Length of latus rectum: 64/3

Exercise: EXERCISE 10.4 | Q: 4 | (Page 27)
For More Understanding

Explanation

This question asks to find the properties of a hyperbola given in general form. The student needs to convert the equation to standard form by dividing by 576, identify a² and b², then apply the standard formulas for hyperbola properties.

Solution Steps

  1. Step 1: Convert to standard form by dividing both sides by 576: 16x²/576 - 9y²/576 = 1, which gives x²/36 - y²/64 = 1

  2. Step 2: Identify a² = 36 and b² = 64, so a = 6 and b = 8

  3. Step 3: Find c using c² = a² + b² = 36 + 64 = 100, so c = 10

  4. Step 4: Vertices are at (±a, 0) = (±6, 0)

  5. Step 5: Foci are at (±c, 0) = (±10, 0)

  6. Step 6: Eccentricity e = c/a = 10/6 = 5/3

  7. Step 7: Length of latus rectum = 2b²/a = 2(64)/6 = 64/3

PreviousNext