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 will follow these instructions:

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

### Step 2: Find the slope of the line
The slope of the line \( y = -5x + 18 \) is \( -5 \). The slope of the line perpendicular to this line will be the negative reciprocal:
\[ \text{slope} = \frac{1}{5} \]

### Step 3: Write the equation of the perpendicular line
Using the point \( C(12, 10) \) and the slope \( \frac{1}{5} \), we can write the equation of the line passing through point \( C \):
\[ y - 10 = \frac{1}{5}(x - 12) \]
Simplifying this equation:
\[ y - 10 = \frac{1}{5}x - \frac{12}{5} \]
\[ y = \frac{1}{5}x - \frac{12}{5} + 10 \]
Converting \( 10 \) to a fraction:
\[ 10 = \frac{50}{5} \]
So,
\[ y = \frac{1}{5}x + \frac{38}{5} \]

### Step 4: Set the two equations equal to find the intersection
Now we will set the equations of the two lines equal to find their intersection:
1. The line of the mouse: \( y = -5x + 18 \)
2. The perpendicular line: \( y = \frac{1}{5}x + \frac{38}{5} \)

Setting them equal:
\[ -5x + 18 = \frac{1}{5}x + \frac{38}{5} \]

### Step 5: Clear the fractions
To eliminate the fraction, multiply the entire equation by \( 5 \):
\[ -25x + 90 = x + 38 \]

### Step 6: Solve for \( x \)
Rearranging gives:
\[ -25x - x = 38 - 90 \]
\[ -26x = -52 \]
\[ x = 2 \]

### Step 7: Substitute \( x \) back to find \( y \)
Now substitute \( x = 2 \) back into either line equation. We will use the mouse's line:
\[ y = -5(2) + 18 \]
\[ y = -10 + 18 = 8 \]

### Step 8: Identify the point \( (a, b) \)
The point where the mouse starts getting farther from the cheese is \( (2, 8) \).

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

### Final Answer
The value of \( a + b \) is \( \boxed{10} \).

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