Chapter 9: Straight Lines

Coordinate geometry is the bridge between algebra and geometry. Once we attach $(x, y)$ to a point, every line, curve, and shape becomes an equation, and every geometric question becomes a question about solving (or manipulating) equations. The straight line is the simplest curve, and we will study it in extraordinary detail.

You have met the slope of a line in earlier classes. Here we make it precise: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$, the change in $y$ per unit change in $x$. Slope is the algebraic encoding of a line's tilt. Equal slopes mean parallel lines; product of slopes equal to $-1$ means perpendicular lines.

Then come the five standard forms of the equation of a line: slope-intercept ($y = mx + c$), point-slope ($y - y_1 = m(x - x_1)$), two-point, intercept ($\dfrac{x}{a} + \dfrac{y}{b} = 1$), and normal ($x \cos\alpha + y \sin\alpha = p$). Knowing which form to pick is half the skill , the right form makes a problem trivial; the wrong form turns it into a swamp.

Distance formulas , between two points, from a point to a line, between parallel lines , are the workhorses of computation. The angle between two lines uses $\tan\theta = \left|\dfrac{m_1 - m_2}{1 + m_1 m_2}\right|$. Solving simultaneous linear equations gives the point of intersection of two lines.

For Class XII, JEE, and beyond, the techniques you build here scale up to circles, conics, and three-dimensional geometry. The mental habit of translating between picture and equation is the most valuable thing this chapter teaches.

What's inside

1. Distance, section formula, and slope , measuring positions and tilts.
2. Various forms of the equation of a line , slope-intercept, point-slope, two-point, intercept, normal.
3. General form Ax + By + C = 0 , converting between forms.
4. Distance of a point from a line; distance between parallel lines.
5. Angle between two lines; parallel and perpendicular conditions.
6. Family of lines and intersection problems.

Key results / Formula card

How to read this chapter

Sketch every problem. A rough diagram, even on a corner of the page, prevents 90% of mistakes. Memorise the five forms but always pick the one that matches the given data , if you know slope and a point, use point-slope; if you know two intercepts, use intercept form.