The task involves writing the equation of a line passing through a given point $(x_0, y_0)$ and forming an angle $\alpha$ with the $Ox$-axis. Let me break this into the steps needed to solve each part.

General Form of the Equation

The slope $k$ of the line is: $k = \tan(\alpha)$ The point-slope form of the line is: $y - y_0 = k(x - x_0)$

Problem 1:

Given:

• $\alpha = \frac{\pi}{4}$,
• $(x_0, y_0) = (-2, 1)$.

1. Compute the slope: $k = \tan\left(\frac{\pi}{4}\right) = 1$

2. Substitute into the point-slope form: $y - 1 = 1(x - (-2))$

3. Simplify: $y - 1 = x + 2 \quad \Rightarrow \quad y = x + 3$

Equation of the line: $y = x + 3$

Problem 2:

Given:

• $\alpha = \arctan(-2)$,
• $(x_0, y_0) = (3, 2)$.

1. Compute the slope: $k = \tan(\arctan(-2)) = -2$

2. Substitute into the point-slope form: $y - 2 = -2(x - 3)$

3. Simplify: $y - 2 = -2x + 6 \quad \Rightarrow \quad y = -2x + 8$

Equation of the line: $y = -2x + 8$

Would you like a step-by-step explanation for a specific part, or help with a similar problem?

Related Questions:

1. How do you derive the point-slope form of a line equation?
2. What is the significance of the slope and how does it relate to the angle $\alpha$?
3. How can $\arctan$ values affect the direction of the line?
4. How do you verify if a point lies on the given line equation?
5. What happens to the line's equation if $\alpha = \frac{\pi}{2}$?

Tip: Always visualize the line's slope to understand the relationship between the angle $\alpha$ and the direction of the line.