The equation of the tangent to the hyperbola `3x^(2) - 4y^(2) = 12`, which makes equal intercepts on the axes is

Options

A. `x = y + 1 = 0`
B. `x + y + 1 = 0`
C. `x + y - 1 = 0`
D. All are correct

Answer

To find the equation of the tangent to the hyperbola \(3x^2 - 4y^2 = 12\) that makes equal intercepts on the axes, we can follow these steps:

### Step 1: Rewrite the Hyperbola Equation
First, we rewrite the equation of the hyperbola in standard form. The given equation is:
\[ 3x^2 - 4y^2 = 12 \]
Dividing the entire equation by 12, we have:
\[ \frac{x^2}{4} - \frac{y^2}{3} = 1 \]
This indicates that \(a^2 = 4\) and \(b^2 = 3\), which gives us \(a = 2\) and \(b = \sqrt{3}\).

### Step 2: Write the Tangent Equation
The general equation of the tangent to the hyperbola in the form \(y = mx \pm \sqrt{a^2m^2 - b^2}\) can be used. Substituting the values of \(a\) and \(b\):
\[ y = mx \pm \sqrt{4m^2 - 3} \]

### Step 3: Equal Intercepts Condition
For the tangent to make equal intercepts on the axes, the x-intercept and y-intercept must be equal. The x-intercept occurs when \(y = 0\):
\[ 0 = mx + \sqrt{4m^2 - 3} \implies x = -\frac{\sqrt{4m^2 - 3}}{m} \]
The y-intercept occurs when \(x = 0\):
\[ y = \pm \sqrt{4m^2 - 3} \]
Setting the absolute values of the intercepts equal, we have:
\[ \left|-\frac{\sqrt{4m^2 - 3}}{m}\right| = \left|\sqrt{4m^2 - 3}\right| \]
This simplifies to:
\[ \frac{\sqrt{4m^2 - 3}}{m} = \sqrt{4m^2 - 3} \]

### Step 4: Solve for \(m\)
Squaring both sides gives:
\[ \frac{4m^2 - 3}{m^2} = 4m^2 - 3 \]
Cross-multiplying leads to:
\[ 4m^2 - 3 = m^2(4m^2 - 3) \implies 4m^2 - 3 = 4m^4 - 3m^2 \]
Rearranging gives us:
\[ 4m^4 - 7m^2 + 3 = 0 \]
Letting \(u = m^2\), we have a quadratic equation:
\[ 4u^2 - 7u + 3 = 0 \]

### Step 5: Use the Quadratic Formula
Applying the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ u = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 4 \cdot 3}}{2 \cdot 4} = \frac{7 \pm \sqrt{49 - 48}}{8} = \frac{7 \pm 1}{8} \]
This gives us:
\[ u = 1 \quad \text{or} \quad u = \frac{3}{4} \]
Thus, \(m^2 = 1\) or \(m^2 = \frac{3}{4}\).

### Step 6: Find Values of \(m\)
Taking square roots, we find:
\[ m = \pm 1 \quad \text{or} \quad m = \pm \frac{\sqrt{3}}{2} \]

### Step 7: Write the Tangent Equations
Substituting these values back into the tangent equation:
1. For \(m = 1\):
\[ y = x \pm 1 \]
This gives us the tangents: \(y = x + 1\) and \(y = x - 1\).

2. For \(m = \frac{\sqrt{3}}{2}\):
\[ y = \frac{\sqrt{3}}{2}x \pm 0 \]
This gives us the tangents: \(y = \frac{\sqrt{3}}{2}x\) and \(y = -\frac{\sqrt{3}}{2}x\).

### Final Answer
The equations of the tangents to the hyperbola that make equal intercepts on the axes are:
\[ y = x + 1, \quad y = x - 1, \quad y = \frac{\sqrt{3}}{2}x, \quad y = -\frac{\sqrt{3}}{2}x \]