The vertices of a triangle are `[at_(1)t_(2),a(t_(1)+t_(2))],[at_(2),a(t_(2)+t_(3))]` and `[at_(3)t_(1),a(t_(3)+t_(1))]`. The co ordinates of the orthocentre of the triangle are…….

Answer

To find the coordinates of the orthocenter of the triangle with vertices given by the coordinates \((at_1t_2, a(t_1 + t_2))\), \((at_2, a(t_2 + t_3))\), and \((at_3t_1, a(t_3 + t_1))\), we will follow these steps:

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are:
- \( A(at_1t_2, a(t_1 + t_2)) \)
- \( B(at_2, a(t_2 + t_3)) \)
- \( C(at_3t_1, a(t_3 + t_1)) \)

### Step 2: Calculate the slopes of the sides
1. **Slope of side BC**:
\[ m_{BC} = \frac{a(t_3 + t_1) - a(t_2 + t_3)}{at_3t_1 - at_2} = \frac{a(t_1 - t_2)}{a(t_3t_1 - t_2)} \]
Simplifying gives:
\[ m_{BC} = \frac{t_1 - t_2}{t_3t_1 - t_2} \]

2. **Slope of side AC**:
\[ m_{AC} = \frac{a(t_3 + t_1) - a(t_1 + t_2)}{at_3t_1 - at_1t_2} = \frac{a(t_3 - t_2)}{a(t_3t_1 - t_1t_2)} = \frac{t_3 - t_2}{t_3t_1 - t_1t_2} \]

3. **Slope of side AB**:
\[ m_{AB} = \frac{a(t_2 + t_3) - a(t_1 + t_2)}{at_2 - at_1t_2} = \frac{a(t_3 - t_1)}{a(t_2 - t_1t_2)} = \frac{t_3 - t_1}{t_2 - t_1t_2} \]

### Step 3: Find the equations of the altitudes
1. **Altitude from A to BC**:
The slope of the altitude from A is the negative reciprocal of \(m_{BC}\):
\[ m_{AD} = -\frac{t_3t_1 - t_2}{t_1 - t_2} \]
Using point-slope form, the equation is:
\[ y - a(t_1 + t_2) = -\frac{t_3t_1 - t_2}{t_1 - t_2}(x - at_1t_2) \]

2. **Altitude from B to AC**:
The slope of the altitude from B is the negative reciprocal of \(m_{AC}\):
\[ m_{BE} = -\frac{t_3t_1 - t_1t_2}{t_3 - t_2} \]
The equation is:
\[ y - a(t_2 + t_3) = -\frac{t_3t_1 - t_1t_2}{t_3 - t_2}(x - at_2) \]

### Step 4: Solve the equations of the altitudes
To find the orthocenter, we need to solve the two equations obtained from the altitudes.

1. Substitute the \(y\) from one altitude equation into the other.
2. Solve for \(x\).
3. Substitute \(x\) back into one of the altitude equations to find \(y\).

### Step 5: Final coordinates of the orthocenter
After solving the equations, we find:
\[ x = -a \]
Substituting \(x = -a\) into the altitude equation gives:
\[ y = -a(t_1 + t_2 + t_3 + t_1t_2t_3) \]

Thus, the coordinates of the orthocenter are:
\[ \text{Orthocenter} = (-a, -a(t_1 + t_2 + t_3 + t_1t_2t_3)) \]