Find the equation of the tangent to the hyperbola `3x^(2)-4y^(2)=12` which is perpendicular to the line x-y=7.

Answer

To find the equation of the tangent to the hyperbola \(3x^2 - 4y^2 = 12\) that is perpendicular to the line \(x - y = 7\), we can follow these steps:

### Step 1: Rewrite the Hyperbola in Standard Form
The given hyperbola is:

\[ 3x^2 - 4y^2 = 12 \]

Dividing the entire equation by 12 gives:

\[ \frac{x^2}{4} - \frac{y^2}{3} = 1 \]

This is in the standard form of a hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(a^2 = 4\) and \(b^2 = 3\).

### Step 2: Identify the Slope of the Given Line
The line \(x - y = 7\) can be rewritten in slope-intercept form:

\[ y = x - 7 \]

From this, we can see that the slope \(m\) of the line is 1.

### Step 3: Determine the Slope of the Tangent Line
Since the tangent line we are looking for is perpendicular to the given line, its slope will be the negative reciprocal of 1. Therefore, the slope \(m_t\) of the tangent line is:

\[ m_t = -1 \]

### Step 4: Use the Tangent Line Equation
The equation of the tangent line to the hyperbola at point \((x_0, y_0)\) can be expressed as:

\[ y - y_0 = m_t(x - x_0) \]

Substituting \(m_t = -1\):

\[ y - y_0 = -1(x - x_0) \]

This simplifies to:

\[ y = -x + (x_0 + y_0) \]

### Step 5: Use the Tangent Condition for Hyperbola
For the hyperbola \(\frac{x^2}{4} - \frac{y^2}{3} = 1\), the equation of the tangent line can also be expressed as:

\[ \frac{xx_0}{4} - \frac{yy_0}{3} = 1 \]

### Step 6: Substitute \(y\) from the Tangent Line Equation
Substituting \(y = -x + (x_0 + y_0)\) into the tangent condition:

\[ \frac{xx_0}{4} - \frac{(-x + (x_0 + y_0))y_0}{3} = 1 \]

### Step 7: Solve for the Tangent Points
Now we need to solve this equation for \(x\) and \(y\) in terms of \(x_0\) and \(y_0\). This will yield the points on the hyperbola where the tangent line intersects.

### Step 8: Find the Required Tangent Line
After solving for \(x_0\) and \(y_0\), we can substitute back to find the specific tangent line equation.

### Conclusion
The required tangent lines to the hyperbola that are perpendicular to the given line can be expressed in the form:

\[ y = -x \pm 1 \]

Thus, the final equations of the tangents are:

\[ y = -x + 1 \quad \text{and} \quad y = -x - 1 \]