line l has equation y=2x+3 find the distance between l and point W(0,-2)

To find the distance between the line \( l \) with the equation \( y = 2x + 3 \) and the point \( W(0, -2) \), we use the formula for the shortest distance between a point \( (x_1, y_1) \) and a line of the form \( Ax + By + C = 0 \):

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

### Step 1: Rewrite the line equation in standard form
The equation of the line \( y = 2x + 3 \) can be rewritten as:
\[
2x - y + 3 = 0
\]
Here, \( A = 2 \), \( B = -1 \), and \( C = 3 \).

### Step 2: Apply the distance formula
Using the point \( W(0, -2) \), where \( x_1 = 0 \) and \( y_1 = -2 \), plug the values into the formula:
\[
d = \frac{|2(0) - 1(-2) + 3|}{\sqrt{2^2 + (-1)^2}} = \frac{|0 + 2 + 3|}{\sqrt{4 + 1}} = \frac{|5|}{\sqrt{5}} = \frac{5}{\sqrt{5}} = \sqrt{5}
\]

Thus, the distance between the line \( l \) and the point \( W(0, -2) \) is \( \sqrt{5} \).

Would you like further clarification or details?

Here are 5 related questions to expand your understanding:
1. How do you find the distance between two parallel lines?
2. How would the distance formula change if the point were on the line?
3. What is the geometric interpretation of the shortest distance from a point to a line?
4. Can you derive the formula for the distance from a point to a line using vector projections?
5. How would the distance calculation differ if the line equation had a negative slope?

**Tip:** Always make sure the line equation is in standard form \( Ax + By + C = 0 \) before applying the distance formula.

Learn how to calculate the distance between the line y = 2x + 3 and the point W(0, -2) using the distance formula for a point and a line. This detailed step-by-step solution covers essential concepts in analytic geometry and linear equations. Suitable for Grades 10-12 students.

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