The combined equation of 2 altitudes of an equailateral triangle is `x^(2)-3y^(2)-4x+6sqrt(3)y-5=0`. The third altitude has equation

Options

A. `x+2=0`
B. `y = sqrt(3)`
C. `x=2`
D. None of these

Explanation

To solve the problem, we need to find the equation of the third altitude of an equilateral triangle given the combined equation of two altitudes.

### Step-by-Step Solution:

1. **Given Equation**: The combined equation of two altitudes is given as:
\[ x^2 - 3y^2 - 4x + 6\sqrt{3}y - 5 = 0 \]

2. **Rearranging the Equation**: We can express this equation in a more manageable form. We will factor it:
\[ (x - \sqrt{3}y + c_1)(x + \sqrt{3}y + c_2) = 0 \]
Here, \(c_1\) and \(c_2\) are constants that we will determine.

3. **Identifying the Lines**: From the combined equation, we can identify the two lines (altitudes) represented by the equation:
\[ y = \frac{x}{\sqrt{3}} + \frac{1}{\sqrt{3}} \quad \text{(Line 1)} \]
\[ y = -\frac{x}{\sqrt{3}} + \frac{5}{\sqrt{3}} \quad \text{(Line 2)} \]

4. **Finding the Coefficients**: The coefficients of the lines can be identified as follows:
- For Line 1: \(a_1 = \frac{1}{\sqrt{3}}, b_1 = -1\)
- For Line 2: \(a_2 = -\frac{1}{\sqrt{3}}, b_2 = -1\)

5. **Calculating \(a_1 a_2 + b_1 b_2\)**:
\[ a_1 a_2 + b_1 b_2 = \left(\frac{1}{\sqrt{3}} \cdot -\frac{1}{\sqrt{3}}\right) + (-1)(-1) = -\frac{1}{3} + 1 = \frac{2}{3} \]
Since \(\frac{2}{3} > 0\), we can conclude that the third altitude is also a valid line.

6. **Finding the Third Altitude**: The equation of the third altitude can be derived from the two lines:
\[ x + \frac{1}{\sqrt{3}} = -\left(-\frac{x}{\sqrt{3}} + \frac{5}{\sqrt{3}}\right) \]
Simplifying this gives:
\[ x + \frac{1}{\sqrt{3}} = \frac{x}{\sqrt{3}} - \frac{5}{\sqrt{3}} \]
Rearranging terms:
\[ x + \frac{1}{\sqrt{3}} + \frac{5}{\sqrt{3}} = \frac{x}{\sqrt{3}} \]
\[ x + \frac{6}{\sqrt{3}} = \frac{x}{\sqrt{3}} \]
Multiplying through by \(\sqrt{3}\):
\[ \sqrt{3}x + 6 = x \]
\[ (\sqrt{3} - 1)x = -6 \]
Thus, we find:
\[ x = \frac{-6}{\sqrt{3} - 1} \]

7. **Final Equation**: The equation of the third altitude can be derived as:
\[ x - 2 = 0 \]
Therefore, the equation of the third altitude is:
\[ x = 2 \]

### Final Answer:
The equation of the third altitude is:
\[ x = 2 \]