Calculate the co-ordinates of the foot of the perpendicular from the point(-4, 2) to the line 3x +2y=5. Also find the equation of the smallest circle passing through (-4, 2) and having its centre on the line 3x + 2y=5.

Answer

To solve the problem, we need to follow these steps:

### Step 1: Identify the given point and line
We are given a point \( P(-4, 2) \) and a line defined by the equation \( 3x + 2y = 5 \).

### Step 2: Find the slope of the line
The equation of the line can be rearranged to slope-intercept form \( y = mx + b \):
\[ 2y = -3x + 5 \implies y = -\frac{3}{2}x + \frac{5}{2} \]
Thus, the slope \( m_1 \) of the line is \( -\frac{3}{2} \).

### Step 3: Find the slope of the perpendicular line
The slope \( m_2 \) of the line perpendicular to the given line is the negative reciprocal of \( m_1 \):
\[ m_2 = \frac{2}{3} \]

### Step 4: Write the equation of the perpendicular line
Using the point-slope form of the line equation, the equation of the line passing through point \( P(-4, 2) \) with slope \( \frac{2}{3} \) is:
\[ y - 2 = \frac{2}{3}(x + 4) \]
Simplifying this:
\[ y - 2 = \frac{2}{3}x + \frac{8}{3} \]
\[ y = \frac{2}{3}x + \frac{8}{3} + 2 \]
\[ y = \frac{2}{3}x + \frac{14}{3} \]

### Step 5: Solve the system of equations
Now we have two equations:
1. \( 3x + 2y = 5 \) (the original line)
2. \( y = \frac{2}{3}x + \frac{14}{3} \) (the perpendicular line)

Substituting the second equation into the first:
\[ 3x + 2\left(\frac{2}{3}x + \frac{14}{3}\right) = 5 \]
\[ 3x + \frac{4}{3}x + \frac{28}{3} = 5 \]
To eliminate the fraction, multiply through by 3:
\[ 9x + 4x + 28 = 15 \]
\[ 13x + 28 = 15 \]
\[ 13x = 15 - 28 \]
\[ 13x = -13 \implies x = -1 \]

### Step 6: Find the y-coordinate
Substituting \( x = -1 \) back into the equation of the perpendicular line:
\[ y = \frac{2}{3}(-1) + \frac{14}{3} = -\frac{2}{3} + \frac{14}{3} = \frac{12}{3} = 4 \]
Thus, the coordinates of the foot of the perpendicular are \( (-1, 4) \).

### Step 7: Find the center of the smallest circle
The center of the smallest circle that passes through \( P(-4, 2) \) and lies on the line \( 3x + 2y = 5 \) can be found by determining the midpoint between \( P(-4, 2) \) and the foot of the perpendicular \( (-1, 4) \):
\[ \text{Center} = \left( \frac{-4 + (-1)}{2}, \frac{2 + 4}{2} \right) = \left( \frac{-5}{2}, 3 \right) \]

### Step 8: Find the radius of the circle
The radius \( r \) is the distance from the center \( \left( -\frac{5}{2}, 3 \right) \) to the point \( P(-4, 2) \):
\[ r = \sqrt{\left(-4 + \frac{5}{2}\right)^2 + (2 - 3)^2} = \sqrt{\left(-\frac{3}{2}\right)^2 + (-1)^2} = \sqrt{\frac{9}{4} + 1} = \sqrt{\frac{9}{4} + \frac{4}{4}} = \sqrt{\frac{13}{4}} = \frac{\sqrt{13}}{2} \]

### Step 9: Write the equation of the circle
The equation of the circle with center \( \left( -\frac{5}{2}, 3 \right) \) and radius \( \frac{\sqrt{13}}{2} \) is:
\[ \left( x + \frac{5}{2} \right)^2 + (y - 3)^2 = \left( \frac{\sqrt{13}}{2} \right)^2 \]
\[ \left( x + \frac{5}{2} \right)^2 + (y - 3)^2 = \frac{13}{4} \]

### Final Answers:
- The coordinates of the foot of the perpendicular are \( (-1, 4) \).
- The equation of the smallest circle is:
\[ \left( x + \frac{5}{2} \right)^2 + (y - 3)^2 = \frac{13}{4} \]