Orthocentre of the triangle formed by the lines whose combined equation is `(y ^(2) - 6xy -x ^(2)) ( 4x -y +7) =0` is

Options

A. `(1,2)`
B. `(2,3)`
C. `(0,1)`
D. none of these

Answer

To find the orthocenter of the triangle formed by the lines whose combined equation is \((y^2 - 6xy - x^2)(4x - y + 7) = 0\), we will follow these steps:

### Step 1: Identify the Lines
The combined equation consists of two parts:
1. \(y^2 - 6xy - x^2 = 0\)
2. \(4x - y + 7 = 0\)

### Step 2: Solve the First Equation
We will first solve the quadratic equation \(y^2 - 6xy - x^2 = 0\) for \(y\):
\[ y^2 - 6xy - x^2 = 0 \]
This can be treated as a quadratic in \(y\). Using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -6x\), and \(c = -x^2\):
\[ y = \frac{6x \pm \sqrt{(-6x)^2 - 4(1)(-x^2)}}{2(1)} \]
\[ y = \frac{6x \pm \sqrt{36x^2 + 4x^2}}{2} \]
\[ y = \frac{6x \pm \sqrt{40x^2}}{2} \]
\[ y = \frac{6x \pm 2\sqrt{10}x}{2} \]
\[ y = 3x \pm \sqrt{10}x \]
Thus, the two lines are:
1. \(y = (3 + \sqrt{10})x\)
2. \(y = (3 - \sqrt{10})x\)

### Step 3: Solve the Second Equation
Now, we will solve the second equation \(4x - y + 7 = 0\) for \(y\):
\[ y = 4x + 7 \]

### Step 4: Find the Intersection Points
To find the orthocenter, we need to find the intersection points of the lines:
1. \(y = (3 + \sqrt{10})x\) and \(y = 4x + 7\)
2. \(y = (3 - \sqrt{10})x\) and \(y = 4x + 7\)

**For the first pair:**
Set \((3 + \sqrt{10})x = 4x + 7\):
\[ (3 + \sqrt{10})x - 4x - 7 = 0 \]
\[ (\sqrt{10} - 1)x - 7 = 0 \]
\[ x = \frac{7}{\sqrt{10} - 1} \]
Substituting back to find \(y\):
\[ y = 4\left(\frac{7}{\sqrt{10} - 1}\right) + 7 \]

**For the second pair:**
Set \((3 - \sqrt{10})x = 4x + 7\):
\[ (3 - \sqrt{10})x - 4x - 7 = 0 \]
\[ (-\sqrt{10} - 1)x - 7 = 0 \]
\[ x = \frac{-7}{\sqrt{10} + 1} \]
Substituting back to find \(y\):
\[ y = 4\left(\frac{-7}{\sqrt{10} + 1}\right) + 7 \]

### Step 5: Determine the Orthocenter
The orthocenter of a right triangle formed by two perpendicular lines is the intersection of the altitudes. Since the triangle is formed by the two lines and the line \(4x - y + 7 = 0\), we can find the orthocenter by calculating the intersection of the altitudes.