Orthocentre of the triangle formed by joining the points `(4,1/4),(3,1/3),(2,1/2)` is

Options

A. `(1/24,24)`
B. `(-1/24,-24)`
C. `(-1/24,24)`
D. `(1/24,-24)`

Answer

To find the orthocenter of the triangle formed by the points \( A(4, \frac{1}{4}) \), \( B(3, \frac{1}{3}) \), and \( C(2, \frac{1}{2}) \), we will follow these steps:

### Step 1: Calculate the slopes of the sides of the triangle
1. **Find the slope of side \( AB \)**:
\[ m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{\frac{1}{3} - \frac{1}{4}}{3 - 4} = \frac{\frac{4 - 3}{12}}{-1} = -\frac{1}{12} \]

2. **Find the slope of side \( BC \)**:
\[ m_{BC} = \frac{y_C - y_B}{x_C - x_B} = \frac{\frac{1}{2} - \frac{1}{3}}{2 - 3} = \frac{\frac{3 - 2}{6}}{-1} = -\frac{1}{6} \]

3. **Find the slope of side \( AC \)**:
\[ m_{AC} = \frac{y_C - y_A}{x_C - x_A} = \frac{\frac{1}{2} - \frac{1}{4}}{2 - 4} = \frac{\frac{2 - 1}{4}}{-2} = -\frac{1}{8} \]

### Step 2: Find the slopes of the altitudes
1. **The slope of the altitude from \( C \) to \( AB \)** (perpendicular to \( AB \)):
\[ m_{CF} = -\frac{1}{m_{AB}} = -\frac{1}{-\frac{1}{12}} = 12 \]

2. **The slope of the altitude from \( A \) to \( BC \)** (perpendicular to \( BC \)):
\[ m_{AG} = -\frac{1}{m_{BC}} = -\frac{1}{-\frac{1}{6}} = 6 \]

### Step 3: Write the equations of the altitudes
1. **Equation of altitude \( CF \)** from point \( C(2, \frac{1}{2}) \):
\[ y - \frac{1}{2} = 12(x - 2) \]
Rearranging gives:
\[ y = 12x - 24 + \frac{1}{2} = 12x - \frac{47}{2} \]

2. **Equation of altitude \( AG \)** from point \( A(4, \frac{1}{4}) \):
\[ y - \frac{1}{4} = 6(x - 4) \]
Rearranging gives:
\[ y = 6x - 24 + \frac{1}{4} = 6x - \frac{95}{4} \]

### Step 4: Solve the equations of the altitudes to find the orthocenter
Set the equations equal to each other:
\[ 12x - \frac{47}{2} = 6x - \frac{95}{4} \]
Multiply through by 4 to eliminate fractions:
\[ 48x - 94 = 24x - 95 \]
Rearranging gives:
\[ 24x = -1 \implies x = -\frac{1}{24} \]

Substituting \( x \) back into one of the altitude equations to find \( y \):
\[ y = 12\left(-\frac{1}{24}\right) - \frac{47}{2} = -\frac{1}{2} - \frac{47}{2} = -24 \]

### Final Answer
The orthocenter of the triangle is:
\[ \left(-\frac{1}{24}, -24\right) \]