line l has equation y=-1/3x - 7/3 find the distance between l and point A(6,2)

To find the distance between a point \( A(x_1, y_1) \) and a line given by the equation \( y = mx + b \), we use the distance formula for a point and a line in the form \( Ax + By + C = 0 \), which is:

\[
\text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\]

First, let's rewrite the line equation \( y = -\frac{1}{3}x - \frac{7}{3} \) in the form \( Ax + By + C = 0 \):

1. Rewrite the equation as:
   \[
   -\frac{1}{3}x - y - \frac{7}{3} = 0
   \]
   This gives \( A = -\frac{1}{3} \), \( B = -1 \), and \( C = -\frac{7}{3} \).

2. Substitute the point \( A(6, 2) \) into the distance formula:
   \[
   \text{Distance} = \frac{\left| -\frac{1}{3}(6) - 1(2) - \frac{7}{3} \right|}{\sqrt{\left(-\frac{1}{3}\right)^2 + (-1)^2}}
   \]

3. Simplify the numerator:
   \[
   -\frac{1}{3}(6) - 2 - \frac{7}{3} = -2 - 2 - \frac{7}{3} = -4 - \frac{7}{3}
   \]
   Convert \( -4 \) to a fraction with a denominator of 3:
   \[
   -4 = -\frac{12}{3}
   \]
   So, the expression becomes:
   \[
   -\frac{12}{3} - \frac{7}{3} = -\frac{19}{3}
   \]

4. Simplify the denominator:
   \[
   \sqrt{\left(-\frac{1}{3}\right)^2 + (-1)^2} = \sqrt{\frac{1}{9} + 1} = \sqrt{\frac{1}{9} + \frac{9}{9}} = \sqrt{\frac{10}{9}} = \frac{\sqrt{10}}{3}
   \]

5. The distance formula becomes:
   \[
   \text{Distance} = \frac{\left| -\frac{19}{3} \right|}{\frac{\sqrt{10}}{3}} = \frac{19}{3} \cdot \frac{3}{\sqrt{10}} = \frac{19}{\sqrt{10}}
   \]

6. Rationalize the denominator:
   \[
   \frac{19}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{19\sqrt{10}}{10}
   \]

Therefore, the distance between the point \( A(6, 2) \) and the line \( y = -\frac{1}{3}x - \frac{7}{3} \) is \( \frac{19\sqrt{10}}{10} \).

Would you like to see more details or have any questions?

Here are five related questions:
1. How do you find the distance between two parallel lines?
2. What is the shortest distance between a point and a plane in 3D?
3. How would the distance calculation change if the line equation was vertical?
4. Can you find the point on the line that is closest to point A?
5. How do you convert a line equation from slope-intercept form to standard form?

**Tip:** Always express the line equation in standard form \( Ax + By + C = 0 \) when using the point-to-line distance formula for simplicity.

Line l has the equation y = -1/3x - 7/3. Find the distance between l and point A(6,2).

Learn how to find the distance between a point and a line using the formula |Ax1 + By1 + C| / sqrt(A^2 + B^2). This example calculates the distance from point A(6,2) to the line y = -1/3x - 7/3, with step-by-step simplification of the formula. Suitable for students in Grades 10-12.

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