Let ABC be a triangle with equations of the sides `AB, BC and CA` respectively `x - 2 = 0,y- 5 = 0 and 5x + 2y - 10 = 0.` Then the orthocentre of the triangle lies on the line

Options

A. x - y = 0
B. 3x - y = 1
C. 4x + y = 13
D. x - 2y = 1

Answer

To find the orthocenter of triangle ABC formed by the lines given by the equations \(AB: x - 2 = 0\), \(BC: y - 5 = 0\), and \(CA: 5x + 2y - 10 = 0\), we will follow these steps:

### Step 1: Identify the vertices of the triangle
The equations of the sides give us the vertices of the triangle.

1. **Line AB**: \(x - 2 = 0\) implies \(x = 2\). This is a vertical line.
2. **Line BC**: \(y - 5 = 0\) implies \(y = 5\). This is a horizontal line.
3. **Line CA**: \(5x + 2y - 10 = 0\).

To find the vertices, we need to find the intersection points of these lines.

- **Point A** (intersection of AB and CA):
Substitute \(x = 2\) into \(5x + 2y - 10 = 0\):
\[ 5(2) + 2y - 10 = 0 \implies 10 + 2y - 10 = 0 \implies 2y = 0 \implies y = 0. \]
So, \(A(2, 0)\).

- **Point B** (intersection of AB and BC):
Since \(x = 2\) and \(y = 5\), we have \(B(2, 5)\).

- **Point C** (intersection of BC and CA):
Substitute \(y = 5\) into \(5x + 2y - 10 = 0\):
\[ 5x + 2(5) - 10 = 0 \implies 5x + 10 - 10 = 0 \implies 5x = 0 \implies x = 0. \]
So, \(C(0, 5)\).

### Step 2: Determine the orthocenter
Since triangle ABC has a right angle at vertex B (because line AB is vertical and line BC is horizontal), the orthocenter of a right triangle is located at the vertex where the right angle is formed.

Thus, the orthocenter \(H\) is at point \(B(2, 5)\).

### Step 3: Find the equation of the line on which the orthocenter lies
We need to check which of the given equations satisfies the coordinates of the orthocenter \(H(2, 5)\).

1. **Option A**: \(x - y + 3 = 0\)
\[ 2 - 5 + 3 = 0 \implies 0 = 0 \quad \text{(True)} \]

2. **Option B**: \(3x - y = 1\)
\[ 3(2) - 5 = 1 \implies 6 - 5 = 1 \quad \text{(True)} \]

3. **Option C**: \(2x + y = 13\)
\[ 2(2) + 5 = 13 \implies 4 + 5 = 9 \quad \text{(False)} \]

4. **Option D**: \(2x - y - 10 = 0\)
\[ 2(2) - 5 - 10 = 0 \implies 4 - 5 - 10 = -11 \quad \text{(False)} \]

### Conclusion
The orthocenter of triangle ABC lies on the lines represented by options A and B.