The equation of the line passing through `(-3, 5)` and perpendicular to the line through the points `(1,0) and (-4, 1)` is

Options

A. `5x+y+10=0`
B. ` 3x-y+20 =0`
C. `5x-y -10 =0`
D. ` 5x+y +20 =0`

Answer

To find the equation of the line passing through the point (-3, 5) and perpendicular to the line through the points (1, 0) and (-4, 1), we can follow these steps:

### Step 1: Find the slope of the line through the points (1, 0) and (-4, 1).

The slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the points (1, 0) and (-4, 1):
\[ m = \frac{1 - 0}{-4 - 1} = \frac{1}{-5} = -\frac{1}{5} \]

### Step 2: Determine the slope of the perpendicular line.

If two lines are perpendicular, the product of their slopes is -1. If the slope of the first line is \( m \), then the slope of the perpendicular line \( m' \) is given by:
\[ m' = -\frac{1}{m} \]

Substituting \( m = -\frac{1}{5} \):
\[ m' = -\frac{1}{-\frac{1}{5}} = 5 \]

### Step 3: Use the point-slope form of the equation of a line.

The point-slope form of the equation of a line is:
\[ y - y_1 = m'(x - x_1) \]
where \( (x_1, y_1) \) is the point the line passes through, and \( m' \) is the slope we just calculated.

Substituting \( (x_1, y_1) = (-3, 5) \) and \( m' = 5 \):
\[ y - 5 = 5(x + 3) \]

### Step 4: Simplify the equation.

Distributing the 5 on the right side:
\[ y - 5 = 5x + 15 \]

Adding 5 to both sides:
\[ y = 5x + 20 \]

### Step 5: Rearranging to standard form.

To express the equation in standard form \( Ax + By + C = 0 \):
\[ 5x - y + 20 = 0 \]

Thus, the equation of the line is:
\[ 5x - y + 20 = 0 \]

### Final Answer:
The equation of the line passing through (-3, 5) and perpendicular to the line through (1, 0) and (-4, 1) is:
\[ 5x - y + 20 = 0 \]

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