An equation of a tangent to the hyperbola `16x^2-25y^2-96x + 100y-356-0`, which makes an angle `pi/4` with the transverse axis is

Options

A. `y=x+2`
B. `y=x-5`
C. `y=x+3`
D. `y=x-2`

Answer

To find the equation of a tangent to the hyperbola given by the equation \(16x^2 - 25y^2 - 96x + 100y - 356 = 0\) that makes an angle of \(\frac{\pi}{4}\) with the transverse axis, we can follow these steps:

### Step 1: Rewrite the Hyperbola Equation
We start with the equation of the hyperbola:
\[ 16x^2 - 25y^2 - 96x + 100y - 356 = 0 \]
We will rearrange it to identify the center and the standard form of the hyperbola.

### Step 2: Complete the Square
Group the \(x\) and \(y\) terms:
\[ 16(x^2 - 6x) - 25(y^2 - 4y) = 356 \]
Now, complete the square for \(x\) and \(y\):
- For \(x^2 - 6x\), add and subtract \(9\) (which is \((\frac{6}{2})^2\)):
\[ x^2 - 6x = (x - 3)^2 - 9 \]
- For \(y^2 - 4y\), add and subtract \(4\) (which is \((\frac{4}{2})^2\)):
\[ y^2 - 4y = (y - 2)^2 - 4 \]
Substituting back, we have:
\[ 16((x - 3)^2 - 9) - 25((y - 2)^2 - 4) = 356 \]
This simplifies to:
\[ 16(x - 3)^2 - 144 - 25(y - 2)^2 + 100 = 356 \]
\[ 16(x - 3)^2 - 25(y - 2)^2 - 44 = 356 \]
\[ 16(x - 3)^2 - 25(y - 2)^2 = 400 \]

### Step 3: Standard Form of the Hyperbola
Dividing through by \(400\):
\[ \frac{(x - 3)^2}{25} - \frac{(y - 2)^2}{16} = 1 \]
This is the standard form of the hyperbola, where \(a^2 = 25\) and \(b^2 = 16\). Thus, \(a = 5\) and \(b = 4\).

### Step 4: Determine the Slope of the Tangent
The angle \(\frac{\pi}{4}\) means that the slope \(m\) of the tangent line is:
\[ m = \tan\left(\frac{\pi}{4}\right) = 1 \]

### Step 5: Equation of the Tangent Line
The equation of the tangent to the hyperbola at any point can be expressed as:
\[ y = mx \pm \sqrt{a^2m^2 - b^2} \]
Substituting \(m = 1\), \(a^2 = 25\), and \(b^2 = 16\):
\[ y = x \pm \sqrt{25 \cdot 1^2 - 16} = x \pm \sqrt{25 - 16} = x \pm \sqrt{9} = x \pm 3 \]
Thus, the equations of the tangents are:
\[ y = x + 3 \quad \text{and} \quad y = x - 3 \]

### Step 6: Final Answer
The equation of the tangent to the hyperbola that makes an angle of \(\frac{\pi}{4}\) with the transverse axis is:
\[ y = x + 3 \quad \text{(or)} \quad y = x - 3 \]