Question 54 of 70beginner🔧 ApplyShort Answer2 marks

In each of the Exercises 7 to 15, find the equations of the hyperbola satisfying the given conditions: Vertices (± 2, 0), foci (± 3, 0)

Correct Answer

x24−y25=1\frac{x^2}{4} - \frac{y^2}{5} = 1

Exercise: EXERCISE 10.4 | Q: 7 | (Page 27)
For More Understanding

Explanation

The question asks for the equation of a hyperbola given its vertices and foci. By observing the coordinates of the vertices (± 2, 0) and foci (± 3, 0), we see that both lie on the x-axis. Following the logic in Example 15 of the provided text (which handles the y-axis case), the standard equation form for a hyperbola with its transverse axis on the x-axis is x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. The value 'a' is derived from the vertices, and 'c' from the foci. The relationship b² = c² - a² is used to find the final parameter b².

Solution Steps

  1. Step 1: Orientation: Since the foci are on the x-axis (coordinates (± 3, 0)), the equation of the hyperbola is of the form x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.

  2. Step 2: Identify a and c: Given vertices are (± 2, 0), so a = 2. Given foci are (± 3, 0), so c = 3.

  3. Step 3: Calculate b²: Using the relation c² = a² + b², we get b² = c² - a². Substituting the values, b² = 3² - 2² = 9 - 4 = 5. Thus, b² = 5.

  4. Step 4: Final Equation: Substituting a² = 4 and b² = 5 into the standard form, the equation is x24−y25=1\frac{x^2}{4} - \frac{y^2}{5} = 1.

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