The coordinates of the foot of the perpendicular from (2,3) to the line `3x+4y - 6 = 0 ` are

Options

A. `(-14/25,-27/25)`
B. `(14/15,-17/25)`
C. `(-14/25,17/25)`
D. `(14/25,27/25)`

Answer

To find the coordinates of the foot of the perpendicular from the point (2, 3) to the line given by the equation \(3x + 4y - 6 = 0\), we can use the formula for the foot of the perpendicular from a point to a line.

### Step-by-Step Solution:

1. **Identify the given point and line equation**:
- The point is \(P(2, 3)\).
- The line equation is \(3x + 4y - 6 = 0\).

2. **Rewrite the line equation in the form \(Ax + By + C = 0\)**:
- Here, \(A = 3\), \(B = 4\), and \(C = -6\).

3. **Use the formula for the foot of the perpendicular**:
The coordinates of the foot of the perpendicular from the point \((x_1, y_1)\) to the line \(Ax + By + C = 0\) are given by:
\[ \left( x, y \right) = \left( \frac{B(Bx_1 - Ay_1) - AC}{A^2 + B^2}, \frac{A(Ay_1 - Bx_1) - BC}{A^2 + B^2} \right) \]

4. **Substitute the values into the formula**:
- Here, \(x_1 = 2\), \(y_1 = 3\), \(A = 3\), \(B = 4\), and \(C = -6\).
- Calculate \(A^2 + B^2\):
\[ A^2 + B^2 = 3^2 + 4^2 = 9 + 16 = 25 \]

5. **Calculate the x-coordinate**:
\[ x = \frac{4(4 \cdot 2 - 3 \cdot 3) - 3 \cdot (-6)}{25} \]
\[ = \frac{4(8 - 9) + 18}{25} = \frac{4(-1) + 18}{25} = \frac{-4 + 18}{25} = \frac{14}{25} \]

6. **Calculate the y-coordinate**:
\[ y = \frac{3(3 \cdot 3 - 4 \cdot 2) - 4 \cdot (-6)}{25} \]
\[ = \frac{3(9 - 8) + 24}{25} = \frac{3(1) + 24}{25} = \frac{3 + 24}{25} = \frac{27}{25} \]

7. **Final coordinates of the foot of the perpendicular**:
The coordinates of the foot of the perpendicular from the point (2, 3) to the line \(3x + 4y - 6 = 0\) are:
\[ \left( \frac{14}{25}, \frac{27}{25} \right) \]

### Final Answer:
The coordinates of the foot of the perpendicular are \(\left( \frac{14}{25}, \frac{27}{25} \right)\).