Coordinate Geometry
Coordinate geometry is where algebra and geometry meet: every point on the plane is an ordered pair (x, y), and every line, distance and midpoint becomes something you can compute rather than measure. The GRE keeps this strictly at high-school level — no vectors, no calculus — and the recurring skills are narrow and very learnable: locate points and quadrants, find and interpret slope, write the equation of a line, and apply the distance and midpoint formulas. A quiet but high-value idea is that slope encodes direction: parallel lines share a slope, perpendicular lines have slopes that multiply to −1, and a horizontal line has slope 0 while a vertical line has an undefined slope. The exam loves to test these as Quantitative Comparisons ("Quantity A: the slope of line ℓ; Quantity B: 0"). This chapter covers the plane, slope and intercepts, equations of lines, distance and midpoint, and the basics of parabolas — each with worked examples, the fastest method, and the traps that cost points.
Topics
⚡ GRE shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Read quadrants from signs alone: (+,+) is I, (−,+) is II, (−,−) is III, (+,−) is IV. Many "which quadrant" questions need no arithmetic.
- In y = mx + b the slope and y-intercept are already visible — always rearrange Ax + By = C into this form before reasoning about a line.
- Parallel = equal slopes; perpendicular = slopes multiply to −1 (negative reciprocals). These two facts settle most line-relationship questions instantly.
- The distance formula is just Pythagoras — watch for 3-4-5 and 5-12-13 point pairs and read off 5 or 13 without the calculator.
- The midpoint is the average of the coordinates; a squared term inside distance forgives sign order, but the midpoint does not — add, do not subtract.
- A parabola's opening direction is the sign of a, and its x-intercept count is decided by the discriminant b² − 4ac — no graphing required.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Writing coordinates in the wrong order — (x, y), with x first, always.
- Calling a vertical line's slope zero; it is undefined (horizontal lines are the ones with slope 0).
- Sign errors in slope from subtracting x and y in opposite orders.
- Using the negative reciprocal for parallel lines (they share a slope) or the same slope for perpendicular lines.
- Subtracting instead of adding in the midpoint formula, or averaging only one coordinate.
- Assuming every parabola crosses the x-axis twice — a negative discriminant means it never touches the axis.
📈 GRE exam insight & question patterns
Quantitative Comparison — line ℓ passes through (0, 0) and (3, −6). Quantity A: the slope of ℓ. Quantity B: 0. Which is greater?
Slope = (−6 − 0)/(3 − 0) = −2. Since −2 < 0, Quantity B is greater. A downward line has negative slope — read direction from the sign.
A question asks how many points two graphs share — a line and a parabola. What decides it?
Set the equations equal to form a quadratic and use its discriminant: b² − 4ac > 0 gives two intersection points, = 0 gives one (tangent), < 0 gives none.
Numeric Entry — M(4, 3) is the midpoint of segment AB where A = (1, 1). Enter the x-coordinate of B.
(1 + x)/2 = 4 → x = 7. Midpoint questions frequently run backwards to a missing endpoint.
The GRE gives two lines and asks if they are perpendicular. Fast check?
Multiply their slopes: if the product is −1 they are perpendicular. For y = 2x + 1 and y = −(1/2)x + 4, 2 × (−1/2) = −1, so yes.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Coordinate Geometry when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (5 topics) | 5/5 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 8 cards |
Formula Reference Sheet
Points, distance & midpoint
| Distance between two points | d = √[ (x₂ − x₁)² + (y₂ − y₁)² ] |
|---|---|
| Midpoint | M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ) |
| Slope | m = (y₂ − y₁) / (x₂ − x₁) |
Lines & parabolas
| Slope-intercept form | y = mx + b (b = y-intercept) |
|---|---|
| Parallel lines | same slope: m₁ = m₂ |
| Perpendicular lines | m₁ × m₂ = −1 |
| x-intercept | set y = 0 and solve for x |
| Parabola (vertex form) | y = a(x − h)² + k, vertex (h, k) |