Question

What is the equation of a hyperbola with a = 3 and c = 7 Assume that the transverse axis is horizontal.

Answer 1

`(x^(2))/(9) - (y^(2))/(40)} = 1`

Explanation:

The standard form of the equation of a hyperbola with center (0,0) and horizontal transverse axis is

`(x^(2))/(a^(2)) - (y^(2))/(b^(2)) = 1`

and the distance c between the foci and the y-axis is given by

c² = a² + b²

1. Calculate b

7² = 3² + b²

49 = 9 + b²

40 = b²

b = √40

2. Write the equation

`(x^(2))/(3^(2)) - (y^(2))/((√(40))^(2)) = 1\\\\\mathbf{(x^(2))/(9) - (y^(2))/(40)}} = \mathbf{1}`

The figure shows your hyperbola with a horizontal transverse axis and c = 3.

Answer 2

The equation would be x²/a² - y²/b² =1 , x²/9 - y²/40 = 1 ....

Explanation:

The standard equation of hyperbola with a horizontal transverse axis is:

(x-h)²/a² - (y-k)² /b² = 1

Use Pythagorean theorem to find the value of b.

c² = a²+b²

c= 7

a = 3

Put the value in the equation:

(7)² = (3)² +b²

49= 9+b²

49-9 = b²

40 = b²

Square root both sides:

√40 = √b²

√40 = b

Assume that the center of hyperbola is(0,0)

Thus

(x-0)²/a² - (y-0)²/b² = 1

x²/a² - y²/b² =1

x²/(3)² - y²/(√40)² = 1

x²/9 - y²/40 = 1

Therefore the equation would be x²/a² - y²/b² =1 , x²/9 - y²/40 = 1 ....