The orthocentre of the triangle formed by the lines `xy=0 and x+y=1` , is

Options

A. `(0,0)`
B. `((1)/(2),(1)/(2))`
C. `((1)/(3),(1)/(3))`
D. `((1)/(4),(1)/(4))`

Explanation

To find the orthocentre of the triangle formed by the lines \(xy=0\) and \(x+y=1\), we can follow these steps:

### Step 1: Identify the lines from the equation \(xy=0\)
The equation \(xy=0\) represents two lines:
- \(x=0\) (the y-axis)
- \(y=0\) (the x-axis)

### Step 2: Identify the third line from the equation \(x+y=1\)
The equation \(x+y=1\) represents a straight line that intersects the axes at:
- When \(x=0\), \(y=1\) (point \(A(0,1)\))
- When \(y=0\), \(x=1\) (point \(B(1,0)\))

### Step 3: Determine the vertices of the triangle
The vertices of the triangle formed by the lines \(x=0\), \(y=0\), and \(x+y=1\) are:
- Vertex at the origin \(O(0,0)\) (intersection of \(x=0\) and \(y=0\))
- Vertex at \(A(0,1)\) (intersection of \(x=0\) and \(x+y=1\))
- Vertex at \(B(1,0)\) (intersection of \(y=0\) and \(x+y=1\))

### Step 4: Identify the type of triangle
The triangle formed by these vertices \(O(0,0)\), \(A(0,1)\), and \(B(1,0)\) is a right triangle, with the right angle at the origin \(O(0,0)\).

### Step 5: Find the orthocentre of the triangle
For a right triangle, the orthocentre is located at the vertex where the right angle is formed. Therefore, the orthocentre of the triangle is at the origin \(O(0,0)\).

### Final Answer
The orthocentre of the triangle formed by the lines \(xy=0\) and \(x+y=1\) is \((0,0)\).

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