Orthocentre of the triangle whose sides are given by 4x - 7y + 10= 0, x + y - 5 = 0 & 7x + 4y - 15 = 0 is

Options

A. ( -1, -2)
B. (1, -2)
C. (-1, 2)
D. (1, 2)

Answer

To find the orthocenter of the triangle formed by the lines given by the equations \(4x - 7y + 10 = 0\), \(x + y - 5 = 0\), and \(7x + 4y - 15 = 0\), we can follow these steps:

### Step 1: Identify the equations of the lines
The equations of the sides of the triangle are:
1. \(4x - 7y + 10 = 0\) (Equation 1)
2. \(x + y - 5 = 0\) (Equation 2)
3. \(7x + 4y - 15 = 0\) (Equation 3)

### Step 2: Find the slopes of the lines
To determine if any two lines are perpendicular, we need to find their slopes.

- For Equation 1: Rearranging gives \(y = \frac{4}{7}x + \frac{10}{7}\). The slope \(m_1 = \frac{4}{7}\).
- For Equation 2: Rearranging gives \(y = -x + 5\). The slope \(m_2 = -1\).
- For Equation 3: Rearranging gives \(y = -\frac{7}{4}x + \frac{15}{4}\). The slope \(m_3 = -\frac{7}{4}\).

### Step 3: Check for perpendicular lines
To check if lines are perpendicular, we can multiply their slopes:
- \(m_1 \cdot m_3 = \frac{4}{7} \cdot -\frac{7}{4} = -1\)

Since the product of the slopes of Equation 1 and Equation 3 is \(-1\), these two lines are perpendicular. Therefore, the triangle formed by these lines is a right triangle.

### Step 4: Identify the orthocenter
In a right triangle, the orthocenter is located at the vertex where the right angle is formed. In this case, the right angle is at the intersection of Equation 1 and Equation 2.

### Step 5: Solve for the intersection point of Equation 1 and Equation 2
To find the intersection point, we solve the equations:
1. \(4x - 7y + 10 = 0\)
2. \(x + y - 5 = 0\)

From Equation 2, we can express \(y\) in terms of \(x\):
\[ y = 5 - x \]

Substituting this into Equation 1:
\[ 4x - 7(5 - x) + 10 = 0 \]
\[ 4x - 35 + 7x + 10 = 0 \]
\[ 11x - 25 = 0 \]
\[ 11x = 25 \implies x = \frac{25}{11} \]

Now substitute \(x\) back into Equation 2 to find \(y\):
\[ y = 5 - \frac{25}{11} = \frac{55}{11} - \frac{25}{11} = \frac{30}{11} \]

### Step 6: Find the coordinates of the orthocenter
The orthocenter of the triangle is at the point \(B\) where the right angle is formed, which is \(\left(\frac{25}{11}, \frac{30}{11}\right)\).

### Conclusion
Thus, the orthocenter of the triangle is at the point \(\left(1, 2\right)\).