A piece of cheese is located at (12, 10) in a coordinate plane. A mouse is at (4,-2) and is running up the line ` y= -5x + 18 `. At the point (a, b), the mouse starts getting farther from the cheese rather than closer to it. The value of `(a+b)` is :

Options

A. 6
B. 10
C. 18
D. 14

Answer

To solve the problem step by step, we need to find the point (a, b) on the line \( y = -5x + 18 \) where the mouse is closest to the cheese located at (12, 10).

### Step 1: Identify the line equation and the coordinates of the cheese and mouse
The mouse is running along the line given by the equation:
\[ y = -5x + 18 \]
The cheese is located at the point (12, 10) and the mouse starts at (4, -2).

### Step 2: Express b in terms of a
Since the point (a, b) lies on the line \( y = -5x + 18 \), we can express b in terms of a:
\[ b = -5a + 18 \]

### Step 3: Find the slope of the line connecting the cheese and the point (a, b)
The slope of the line connecting the cheese (12, 10) and the point (a, b) is given by:
\[ \text{slope} = \frac{10 - b}{12 - a} \]

### Step 4: Determine the slope of the perpendicular line
The slope of the line on which the mouse is running is -5. The slope of the line perpendicular to this will be the negative reciprocal:
\[ \text{slope of perpendicular line} = \frac{1}{5} \]

### Step 5: Set up the equation for the slope condition
Since the slope of the line connecting the cheese and the point (a, b) is equal to \(\frac{1}{5}\), we can set up the equation:
\[ \frac{10 - b}{12 - a} = \frac{1}{5} \]

### Step 6: Cross-multiply to solve for a and b
Cross-multiplying gives us:
\[ 5(10 - b) = 12 - a \]
Substituting \( b = -5a + 18 \) into this equation:
\[ 5(10 - (-5a + 18)) = 12 - a \]
This simplifies to:
\[ 5(10 + 5a - 18) = 12 - a \]
\[ 5(5a - 8) = 12 - a \]
\[ 25a - 40 = 12 - a \]

### Step 7: Solve for a
Rearranging gives:
\[ 25a + a = 12 + 40 \]
\[ 26a = 52 \]
\[ a = 2 \]

### Step 8: Solve for b using the value of a
Now substitute \( a = 2 \) back into the equation for b:
\[ b = -5(2) + 18 \]
\[ b = -10 + 18 = 8 \]

### Step 9: Calculate \( a + b \)
Finally, we find \( a + b \):
\[ a + b = 2 + 8 = 10 \]

Thus, the value of \( a + b \) is:
\[ \boxed{10} \]