A hyperbola passes through (2,3) and has asymptotes `3x-4y+5=0` and `12 x+5y-40=0` . Then, the equation of its transverse axis is `77 x-21 y-265=0` `21 x-77 y+265=0` `21 x-77 y-265=0` `21 x+77 y-265=0`

Options

A. `77x-21y-265=0`
B. `21x-77y+265=0`
C. `21x-77y-265=0`
D. `21x+77y-265=0`

Explanation

To solve the problem step by step, we need to find the equation of the transverse axis of the hyperbola given its asymptotes and a point through which it passes.

### Step 1: Identify the equations of the asymptotes
The asymptotes of the hyperbola are given as:
1. \( 3x - 4y + 5 = 0 \)
2. \( 12x + 5y - 40 = 0 \)

### Step 2: Write the equations in slope-intercept form
To find the slopes of the asymptotes, we can rewrite the equations in the form \( y = mx + b \).

1. For the first asymptote:
\[ 3x - 4y + 5 = 0 \implies 4y = 3x + 5 \implies y = \frac{3}{4}x + \frac{5}{4} \]
The slope \( m_1 = \frac{3}{4} \).

2. For the second asymptote:
\[ 12x + 5y - 40 = 0 \implies 5y = -12x + 40 \implies y = -\frac{12}{5}x + 8 \]
The slope \( m_2 = -\frac{12}{5} \).

### Step 3: Find the angle bisectors
The transverse axis of the hyperbola is the angle bisector of the two asymptotes. The equation of the angle bisector can be derived from the asymptotes using the formula:
\[ \frac{3x - 4y + 5}{\sqrt{3^2 + (-4)^2}} = \frac{12x + 5y - 40}{\sqrt{12^2 + 5^2}} \]

Calculating the denominators:
- For the first asymptote:
\[ \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

- For the second asymptote:
\[ \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Step 4: Set up the equation
Now substituting the values into the angle bisector equation:
\[ \frac{3x - 4y + 5}{5} = \frac{12x + 5y - 40}{13} \]

### Step 5: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 13(3x - 4y + 5) = 5(12x + 5y - 40) \]

Expanding both sides:
\[ 39x - 52y + 65 = 60x + 25y - 200 \]

### Step 6: Rearranging the equation
Rearranging the equation to one side:
\[ 39x - 60x - 52y - 25y + 65 + 200 = 0 \]
\[ -21x - 77y + 265 = 0 \]

### Step 7: Final equation
Multiplying through by -1 to make the coefficients positive:
\[ 21x + 77y - 265 = 0 \]

### Conclusion
The equation of the transverse axis is:
\[ 21x + 77y - 265 = 0 \]