The foot of the perpendicular drawn from the point (7, 8) to the line 2x + 3y – 4 = 0 is -

Options

A. `(23/13, 2/13)`
B. `(13, 23/13)`
C. `( - 23/13 , - 2/13)`
D. `( - 2/13 , 23/13)`

Answer

To find the foot of the perpendicular drawn from the point (7, 8) to the line given by the equation \(2x + 3y - 4 = 0\), we can use the formula for the foot of the perpendicular from a point to a line.

### Step-by-Step Solution:

1. **Identify the coefficients of the line equation**:
The line equation is \(2x + 3y - 4 = 0\).
Here, \(a = 2\), \(b = 3\), and \(c = -4\).

2. **Identify the coordinates of the point**:
The point from which we are drawing the perpendicular is \(P(7, 8)\).
Thus, \(x_1 = 7\) and \(y_1 = 8\).

3. **Use the formula for the foot of the perpendicular**:
The coordinates of the foot of the perpendicular \((x, y)\) can be found using the following relationships:
\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = -\frac{ax_1 + by_1 + c}{a^2 + b^2} \]
Substituting the values we have:
\[ \frac{x - 7}{2} = \frac{y - 8}{3} = -\frac{2(7) + 3(8) - 4}{2^2 + 3^2} \]

4. **Calculate the right-hand side**:
First, calculate \(2(7) + 3(8) - 4\):
\[ 14 + 24 - 4 = 34 \]
Now calculate \(a^2 + b^2\):
\[ 2^2 + 3^2 = 4 + 9 = 13 \]
Thus, we have:
\[ -\frac{34}{13} \]

5. **Set up the equations**:
Now we can set up the equations:
\[ \frac{x - 7}{2} = \frac{y - 8}{3} = -\frac{34}{13} \]

6. **Solve for \(x\)**:
From \(\frac{x - 7}{2} = -\frac{34}{13}\):
\[ x - 7 = 2 \left(-\frac{34}{13}\right) \]
\[ x - 7 = -\frac{68}{13} \]
\[ x = 7 - \frac{68}{13} = \frac{91}{13} - \frac{68}{13} = \frac{23}{13} \]

7. **Solve for \(y\)**:
From \(\frac{y - 8}{3} = -\frac{34}{13}\):
\[ y - 8 = 3 \left(-\frac{34}{13}\right) \]
\[ y - 8 = -\frac{102}{13} \]
\[ y = 8 - \frac{102}{13} = \frac{104}{13} - \frac{102}{13} = \frac{2}{13} \]

8. **Final coordinates of the foot of the perpendicular**:
Thus, the foot of the perpendicular from the point (7, 8) to the line \(2x + 3y - 4 = 0\) is:
\[ \left(\frac{23}{13}, \frac{2}{13}\right) \]

### Final Answer:
The foot of the perpendicular is \(\left(\frac{23}{13}, \frac{2}{13}\right)\).