The coordinats of the orthocentre of the triangle whose sides lie along the lines `x=0,y=0` and `x+y-1=0` is
(i) (0,0)
(ii) `(0,-(8)/(3))`
(iii) `(-2,0)`
(iv) `(1/2,1/2)`

Options

A. (0,0)
B. `(0,-(8)/(3))`
C. `(-2,0)`
D. `(1/2,1/2)`

Answer

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

### Step 1: Identify the lines and their intersections
The lines given are:
1. \(x = 0\) (the y-axis)
2. \(y = 0\) (the x-axis)
3. \(x + y - 1 = 0\) (which can be rewritten as \(y = 1 - x\))

### Step 2: Find the vertices of the triangle
To find the vertices of the triangle formed by these lines, we need to find the points of intersection of these lines.

1. **Intersection of \(x = 0\) and \(y = 0\)**:
- This gives the point \((0, 0)\).

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

3. **Intersection of \(y = 0\) and \(x + y - 1 = 0\)**:
- Substituting \(y = 0\) into \(x + y - 1 = 0\) gives:
\[ x + 0 - 1 = 0 \implies x = 1 \]
- This gives the point \((1, 0)\).

Thus, the vertices of the triangle are \((0, 0)\), \((0, 1)\), and \((1, 0)\).

### Step 3: Identify the type of triangle
The triangle formed by these points is a right triangle, with the right angle at the vertex \((0, 0)\).

### Step 4: Find the orthocenter
For a right triangle, the orthocenter is located at the vertex where the right angle is formed. In this case, since the right angle is at \((0, 0)\), the orthocenter is also at this point.

### Conclusion
Thus, the coordinates of the orthocenter of the triangle are:
\[ \text{Orthocenter} = (0, 0) \]

### Final Answer
The correct option is (i) \((0, 0)\).

---