Q.Reduce the equation $\sqrt{3}x + y + 2 = 0$ to the normal form $x\cos\alpha + y\sin\alpha = p$ and hence find the value of $\alpha$ and $p$.

Various Forms of the Equation of a Line

Imagine you want to describe a straight line to someone who has never seen it. You could say "it goes through this point and slants like this" — that's the intuition. In coordinate geometry, we capture that same idea using equations. A line is just the set of all points $(x, y)$ that satisfy a certain condition. Different conditions give us different forms of the same line.

1. Slope-Intercept Form: $y = mx + c$

This is the most familiar form. Here $m$ is the slope (steepness) and $c$ is the y-intercept (where the line cuts the y-axis).

Why it works: If you know how much the line rises for every unit it runs horizontally ($m$), and where it starts on the y-axis ($c$), you can write the equation directly.

Example: A line with slope 2 and y-intercept -3 is $y = 2x - 3$.

2. Point-Slope Form: $y - y_1 = m(x - x_1)$

Suppose you know the slope $m$ and one point $(x_1, y_1)$ on the line. The point-slope form says: the difference in y from that point equals the slope times the difference in x.

Intuition: If you stand at $(x_1, y_1)$ and move horizontally by $(x - x_1)$, you must move vertically by $m$ times that amount to stay on the line.

Example: Line through $(2, 5)$ with slope $-4$: $y - 5 = -4(x - 2)$.

3. Two-Point Form: $\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}$

If you know two points $(x_1, y_1)$ and $(x_2, y_2)$, you don't need to compute slope separately. This form says the ratio of vertical change to total vertical span equals the ratio of horizontal change to total horizontal span.

Why it's natural: It's just the condition that the three points $(x_1, y_1)$, $(x_2, y_2)$, and $(x, y)$ are collinear — they all lie on the same straight line.

Example: Line through $(1, 2)$ and $(3, 8)$: $\frac{y - 2}{8 - 2} = \frac{x - 1}{3 - 1}$, which simplifies to $y = 3x - 1$.

4. Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$

Here $a$ is the x-intercept (where the line cuts the x-axis) and $b$ is the y-intercept.

Intuition: When $y = 0$, the equation gives $x = a$; when $x = 0$, it gives $y = b$. So the line passes through $(a, 0)$ and $(0, b)$.

Example: A line with x-intercept 4 and y-intercept -3: $\frac{x}{4} + \frac{y}{-3} = 1$, or $\frac{x}{4} - \frac{y}{3} = 1$.

5. Normal Form: $x \cos \theta + y \sin \theta = p$

This is the most geometric form. Here $p$ is the perpendicular distance from the origin to the line, and $\theta$ is the angle that this perpendicular makes with the positive x-axis.

Why it's useful: It directly gives the distance of the line from the origin — something the other forms hide.