A hyperbola touches y axis and has its centre at (5/2,20)and one of the focii at (10/24) respectively , find length of the transverse axis

To find the length of the transverse axis of the hyperbola, we first need to understand the key components of its equation based on the information provided. Since the hyperbola touches the y-axis and has its center at (5/2, 20), we can infer a few important details that will help us derive the equation of the hyperbola and ultimately find the length of the transverse axis.

Identifying the Basic Structure

The general equation of a hyperbola that opens horizontally is given by the form:

(y - k)²/a² - (x - h)²/b² = 1

Where (h, k) is the center, a is the distance from the center to the vertices (half the length of the transverse axis), and b is the distance related to the conjugate axis. In our case, the center is at (5/2, 20), which means h = 5/2 and k = 20.

Determining the Orientation

Since the hyperbola touches the y-axis, it indicates that the distance from the center of the hyperbola to the y-axis is equal to a. The distance from the center (5/2, 20) to the y-axis (which is x = 0) is 5/2. Therefore, we can conclude:

a = 5/2

Finding the Focal Distance

Next, we need to determine the distance to the foci, denoted as c. The foci are at a distance of c from the center along the transverse axis. We know one of the foci is located at (10, 24). The distance from the center (5/2, 20) to the focus (10, 24) can be calculated using the distance formula:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Plugging in the coordinates:

Distance = √[(10 - 5/2)² + (24 - 20)²]

= √[(20/2 - 5/2)² + (4)²]

= √[(15/2)² + 4²]

= √[(225/4) + 16]

= √[(225/4) + (64/4)] = √(289/4) = 17/2

Calculating the Length of the Transverse Axis

In a hyperbola, the relationship between a, b, and c is given by:

c² = a² + b²

We already have a = 5/2, and we calculated c = 17/2. Now, let's square these values:

• a² = (5/2)² = 25/4
• c² = (17/2)² = 289/4

Substituting these values into the relationship:

289/4 = 25/4 + b²

To isolate b², we subtract 25/4 from both sides:

b² = (289/4 - 25/4) = 264/4 = 66

Final Step: Length of the Transverse Axis

Now that we have a, we can find the length of the transverse axis, which is given by 2a:

Length of the transverse axis = 2a = 2 * (5/2) = 5

Therefore, the length of the transverse axis of the hyperbola is 5.