Q.For specifying a straight line, how many geometrical parameters should be known?
(A) $1$
(B) $2$
(C) $4$
(D) $3$

Concept: Degrees of freedom for a straight line in the plane.

A straight line in two-dimensional geometry has exactly two independent parameters that uniquely determine it.

Think of the most common forms:

• Slope-intercept form $y = mx + c$ requires two parameters: slope $m$ and $y$-intercept $c$.
• Intercept form $\frac{x}{a} + \frac{y}{b} = 1$ requires two intercepts: $a$ and $b$.
• Normal form $x\cos\theta + y\sin\theta = p$ requires two parameters: the angle $\theta$ of the perpendicular from origin and the perpendicular distance $p$. …

Why Two Parameters?

Think about what it means to "specify" a line: you need enough information to distinguish it from every other possible line in the plane. A line has infinitely many points, but it's constrained by its direction and position.

The key insight is that a line has one degree of freedom in direction (which way it points) and one degree of freedom in position (where it sits in the plane). Together, these give us two independent parameters.

Let me show you why this makes sense through different representations:

Different Ways to See the Two Parameters

1. Slope-intercept form: $y = mx + c$

The two parameters are obvious here:

• $m$ (slope) tells you the direction/steepness
• $c$ (y-intercept) tells you where it crosses the y-axis

2. Two-point form

To draw a unique line, you need exactly two distinct points. One point isn't enough (infinitely many lines pass through a single point), but two points determine the line completely. Since each point in 2D requires 2 coordinates, you might think that's 4 parameters — but there's a constraint: the points must be distinct and the line's equation has only 2 free parameters after we account for the relationship between the points.

3. Normal form: $x \cos\theta + y \sin\theta = p$

Here the two parameters are:

• $\theta$ (angle the perpendicular makes with the x-axis) — direction
• $p$ (perpendicular distance from origin) — position

4. General form: $ax + by + c = 0$ …