Distance Calculation: Origin to Line

The problem asks for the distance between the origin $(0,0)$ and the line defined by the equation $4x + 3y + 6 = 0$.

Distance Formula Application

We use the formula for the perpendicular distance from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$, which is:

$ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} $

Applying Values

For this specific problem:

• The point $(x_0, y_0)$ is the origin $(0, 0)$.
• The line equation is $4x + 3y + 6 = 0$, so $A=4$, $B=3$, and $C=6$.

Step-by-Step Calculation

1. Substitute the values into the distance formula:

   $ d = \frac{|4(0) + 3(0) + 6|}{\sqrt{4^2 + 3^2}} $

2. Simplify the numerator:

   $ |4(0) + 3(0) + 6| = |0 + 0 + 6| = |6| = 6 $

3. Simplify the denominator:

   $ \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 $

4. Calculate the final distance:

   $ d = \frac{6}{5} $

Conclusion

The distance from the origin to the line $4x + 3y + 6 = 0$ is $\frac{6}{5}$.