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

Options

A. `(-1, 1)`
B. `(0, 0)`
C. `(1, 0)`
D. `(0, 1)`

Answer

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 equation \(x + y = 1\) represents a straight line that intersects the axes.

### Step 2: Find the points of intersection
Next, we need to find the points where the line \(x + y = 1\) intersects the axes:
1. **Intersection with the y-axis**: Set \(x = 0\):
\[ 0 + y = 1 \implies y = 1 \]
So, the point is \((0, 1)\).

2. **Intersection with the x-axis**: Set \(y = 0\):
\[ x + 0 = 1 \implies x = 1 \]
So, the point is \((1, 0)\).

### Step 3: Identify the vertices of the triangle
The vertices of the triangle formed by the lines \(x = 0\), \(y = 0\), and \(x + y = 1\) are:
- \(A(0, 0)\) (the origin)
- \(B(0, 1)\) (intersection with the y-axis)
- \(C(1, 0)\) (intersection with the x-axis)

### Step 4: Determine the orthocenter
The orthocenter of a triangle is the point where the altitudes intersect. In a right triangle, the orthocenter is located at the vertex where the right angle is formed.

In our triangle:
- The right angle is at point \(A(0, 0)\).

### Conclusion
Thus, the orthocenter of the triangle formed by the lines \(xy = 0\) and \(x + y = 1\) is at the point:
\[ \text{Orthocenter} = (0, 0) \]

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