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

Options

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

Explanation

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

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

The line \(x + y = 1\) can be rewritten as \(y = 1 - x\).

### Step 2: Find the points of intersection
Next, we need to find the points where these lines intersect to form the vertices of the triangle.

1. **Intersection of \(x = 0\) and \(x + y = 1\)**:
- Substitute \(x = 0\) into \(x + y = 1\):
\[ 0 + y = 1 \implies y = 1 \]
- So, the point is \((0, 1)\).

2. **Intersection of \(y = 0\) and \(x + y = 1\)**:
- Substitute \(y = 0\) into \(x + y = 1\):
\[ x + 0 = 1 \implies x = 1 \]
- So, the point is \((1, 0)\).

3. **Intersection of \(x = 0\) and \(y = 0\)**:
- The intersection point is \((0, 0)\).

### Step 3: Identify the vertices of the triangle
The vertices of the triangle formed by the lines are:
- \(A(0, 1)\)
- \(B(1, 0)\)
- \(C(0, 0)\)

### Step 4: Determine if the triangle is a right triangle
We can check if the triangle is a right triangle by examining the angles formed by the lines:
- The angle at point \(C(0, 0)\) is \(90^\circ\) since the lines \(x = 0\) and \(y = 0\) are perpendicular.

### Step 5: Find the orthocenter
In a right triangle, the orthocenter is located at the vertex where the right angle is formed. Since our triangle has a right angle at point \(C(0, 0)\), the orthocenter is at:
\[ \text{Orthocenter} = C(0, 0) \]

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

---