What is the distance of the point (2 , 3) from the line ` 2 x + 3y + 4 = 0 ? `

Options

A. A)`(15)/( sqrt(13))`
B. B)`(16)/( sqrt(13))`
C. C)`(17)/(sqrt(13))`
D. D)`(8)/( sqrt(3))`

Answer

To find the distance of the point (2, 3) from the line given by the equation \(2x + 3y + 4 = 0\), we can use the formula for the distance \(D\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\):

\[ D = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

### Step-by-Step Solution:

1. **Identify the coefficients from the line equation**:
The line equation is \(2x + 3y + 4 = 0\). Here, we can identify:
- \(A = 2\)
- \(B = 3\)
- \(C = 4\)

2. **Identify the coordinates of the point**:
The point given is \((x_0, y_0) = (2, 3)\).

3. **Substitute the values into the distance formula**:
Now, we substitute \(A\), \(B\), \(C\), \(x_0\), and \(y_0\) into the distance formula:
\[ D = \frac{|2(2) + 3(3) + 4|}{\sqrt{2^2 + 3^2}} \]

4. **Calculate the numerator**:
First, calculate the expression inside the absolute value:
\[ 2(2) + 3(3) + 4 = 4 + 9 + 4 = 17 \]
So, the numerator becomes:
\[ |17| = 17 \]

5. **Calculate the denominator**:
Now, calculate the denominator:
\[ \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]

6. **Combine the results**:
Now, we can write the distance:
\[ D = \frac{17}{\sqrt{13}} \]

### Final Result:
The distance of the point (2, 3) from the line \(2x + 3y + 4 = 0\) is:
\[ \frac{17}{\sqrt{13}} \]