Find the co-ordinates of the orthocentre of a triangle whose vertices are `(3,-1)`, `(-1,2)` and `(0,0)`.

Answer

To find the coordinates of the orthocenter of the triangle with vertices A(3, -1), B(-1, 2), and C(0, 0), we will follow these steps:

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are:
- A(3, -1)
- B(-1, 2)
- C(0, 0)

### Step 2: Calculate the slopes of the sides
1. **Slope of line AB (M_AB)**:
\[ M_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-1)}{-1 - 3} = \frac{3}{-4} = -\frac{3}{4} \]

2. **Slope of line BC (M_BC)**:
\[ M_{BC} = \frac{0 - 2}{0 - (-1)} = \frac{-2}{1} = -2 \]

3. **Slope of line CA (M_CA)**:
\[ M_{CA} = \frac{-1 - 0}{3 - 0} = \frac{-1}{3} \]

### Step 3: Find the slopes of the altitudes
1. **Slope of altitude from C to AB (M_C)**:
Since the altitude is perpendicular to AB, we take the negative reciprocal of M_AB:
\[ M_C = \frac{4}{3} \]

2. **Slope of altitude from A to BC (M_A)**:
Since the altitude is perpendicular to BC, we take the negative reciprocal of M_BC:
\[ M_A = \frac{1}{2} \]

### Step 4: Write the equations of the altitudes
1. **Equation of altitude from C (0, 0) to AB**:
Using point-slope form \(y - y_1 = m(x - x_1)\):
\[ y - 0 = \frac{4}{3}(x - 0) \implies y = \frac{4}{3}x \]

2. **Equation of altitude from A (3, -1) to BC**:
Using point-slope form:
\[ y - (-1) = \frac{1}{2}(x - 3) \implies y + 1 = \frac{1}{2}x - \frac{3}{2} \implies y = \frac{1}{2}x - \frac{5}{2} \]

### Step 5: Solve the equations of the altitudes to find the orthocenter
Set the two equations equal to each other:
\[ \frac{4}{3}x = \frac{1}{2}x - \frac{5}{2} \]

To eliminate fractions, multiply through by 6 (the least common multiple of the denominators):
\[ 8x = 3x - 15 \]
\[ 8x - 3x = -15 \implies 5x = -15 \implies x = -3 \]

Now substitute \(x = -3\) back into one of the altitude equations to find \(y\):
Using \(y = \frac{4}{3}x\):
\[ y = \frac{4}{3}(-3) = -4 \]

### Step 6: Conclusion
The coordinates of the orthocenter of the triangle are:
\[ \text{Orthocenter} = (-3, -4) \]