Image of the point `(-1, 3)` with respect to the line y = 2x is

Options

A. (7/5, 14/5)
B. (1, 2)
C. (3, 1)
D. (5, 1)

Answer

To find the image of the point \((-1, 3)\) with respect to the line \(y = 2x\), we can follow these steps:

### Step 1: Identify the point and the line
The given point is \(P(-1, 3)\) and the line is given by the equation \(y = 2x\). We can rewrite the line in standard form as \(2x - y = 0\).

### Step 2: Determine coefficients for the line
From the line equation \(2x - y + 0 = 0\), we identify:
- \(a = 2\)
- \(b = -1\)
- \(c = 0\)

### Step 3: Use the formula for the image of a point
The formula for the image \(P'\) of a point \(P(x_1, y_1)\) with respect to the line \(ax + by + c = 0\) is given by:
\[ \left( x', y' \right) = \left( x_1 - \frac{2a(ax_1 + by_1 + c)}{a^2 + b^2}, y_1 - \frac{2b(ax_1 + by_1 + c)}{a^2 + b^2} \right) \]

### Step 4: Substitute the values
Substituting \(x_1 = -1\), \(y_1 = 3\), \(a = 2\), \(b = -1\), and \(c = 0\) into the formula:

1. Calculate \(ax_1 + by_1 + c\):
\[ ax_1 + by_1 + c = 2(-1) + (-1)(3) + 0 = -2 - 3 + 0 = -5 \]

2. Calculate \(a^2 + b^2\):
\[ a^2 + b^2 = 2^2 + (-1)^2 = 4 + 1 = 5 \]

3. Substitute into the formula:
\[ x' = -1 - \frac{2 \cdot 2 \cdot (-5)}{5} = -1 + \frac{20}{5} = -1 + 4 = 3 \]
\[ y' = 3 - \frac{2 \cdot (-1) \cdot (-5)}{5} = 3 - \frac{10}{5} = 3 - 2 = 1 \]

### Step 5: Conclusion
The image of the point \((-1, 3)\) with respect to the line \(y = 2x\) is \((3, 1)\).