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

Options

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

Answer

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

### Step 1: Identify the lines and their intersections
The line \(x + y = 1\) can be rewritten in slope-intercept form as \(y = -x + 1\). This line intersects the axes at the points:
- When \(x = 0\), \(y = 1\) → Point A (0, 1)
- When \(y = 0\), \(x = 1\) → Point B (1, 0)

The equation \(xy = 0\) represents the x-axis and y-axis. Therefore, the points of intersection with the axes are:
- Point C (0, 0), which is the origin.

### Step 2: Determine the vertices of the triangle
The vertices of the triangle formed by the lines \(x + y = 1\) and \(xy = 0\) are:
- A (0, 1)
- B (1, 0)
- C (0, 0)

### Step 3: Identify the right angle
To find the orthocenter, we need to check if the triangle is a right triangle. The right angle is located at point C (0, 0) since the lines \(x + y = 1\) and the axes (x-axis and y-axis) are perpendicular to each other.

### Step 4: Find the orthocenter
In a right triangle, the orthocenter is located at the vertex where the right angle is formed. Since point C (0, 0) is where the right angle is, the orthocenter of the triangle is at point C.

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

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