The distance of the point `P(1,-3)` from the line `2y-3x=4` is

Options

A. 13
B. `(2)/(sqrt2)`
C. `sqrt13`
D. `1/13`

Answer

To find the distance of the point \( P(1, -3) \) from the line given by the equation \( 2y - 3x = 4 \), we can use the formula for the distance from a point to a line. The standard form of the line equation is \( Ax + By + C = 0 \).

### Step 1: Rewrite the line equation in standard form
The given line equation is:
\[ 2y - 3x = 4 \]
We can rearrange it to the standard form:
\[ -3x + 2y - 4 = 0 \]
This gives us:
\[ A = -3, \quad B = 2, \quad C = -4 \]

### Step 2: Identify the coordinates of the point
The coordinates of the point \( P \) are:
\[ m = 1, \quad n = -3 \]

### Step 3: Use the distance formula
The formula for the distance \( d \) from a point \( (m, n) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Am + Bn + C|}{\sqrt{A^2 + B^2}} \]

### Step 4: Substitute the values into the formula
Substituting the values we have:
\[ d = \frac{|-3(1) + 2(-3) - 4|}{\sqrt{(-3)^2 + (2)^2}} \]

Calculating the numerator:
\[ -3(1) + 2(-3) - 4 = -3 - 6 - 4 = -13 \]
Taking the absolute value:
\[ |-13| = 13 \]

Calculating the denominator:
\[ \sqrt{(-3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

### Step 5: Final calculation of distance
Now substituting back into the distance formula:
\[ d = \frac{13}{\sqrt{13}} = \sqrt{13} \]

Thus, the distance of the point \( P(1, -3) \) from the line \( 2y - 3x = 4 \) is:
\[ \sqrt{13} \]