Question

10
The transverse axis of a hyperbola is of length 2a and a vertex divides the segment of the axis
between the centre and the corresponding focus in the ratio 2 : 1. The equation of the hyperbola is
1) 5x2 - 4y2 = 5a
2) 5x2 - 4y2 = 4a 3) 4x2 -- 5y2 = 5a? 4) 4x² - 5y2 = 4a?​

Answer 1

Given: The transverse axis of a hyperbola is of length 2a and a vertex divides the segment of the axis  between the centre and the corresponding focus in the ratio 2 : 1.

To find: The equation of the hyperbola ?

Solution:

• Now we have given that :

                   a / ae-a = 2/1

• Solving this, we get:

                   a = 2(ae - a)

                   1 = 2e - 2

                   3 = 2e

                   e = 3/2

• Now we know :

                   e² = 1 + b²/a²

                   (3/2)² = 1 + b²/a²

                   b²/a² = 9/4 - 1

                   b²/a² = 5/4

                   b² = 5a²/4

• Now the hyperbola is:

                   x²/a² - y²/b² = 1

• Putting the values, we get:

                   x²/a² - y²/(5a²/4)² = 1

• Solving this, we get:

                   x²/a² - 4y²/5a² = 1

                   5x² - 4y²/5a² = 1

                   5x² - 4y² = 5a²

Answer:

           So the equation of the hyperbola is 5x² - 4y² = 5a².