Coordinate geometry questions require working with lines, slopes, distances, and midpoints on the xy-plane. Classical geometry (triangles, circles, etc.) was removed from GMAT Focus, but coordinate geometry remains and uses algebraic tools.

A) The Coordinate Plane

Points are written as (x, y). The x-axis is horizontal; the y-axis is vertical. They intersect at the origin (0, 0).
Quadrants: I (x>0, y>0), II (x<0, y>0), III (x<0, y<0), IV (x>0, y<0).

B) Slope

Slope m = rise / run = (y₂ − y₁) / (x₂ − x₁). Subtract in the same order for both coordinates.

Positive slope: line goes up left-to-right. Negative slope: line goes down left-to-right.
Zero slope: horizontal line (y = constant). Undefined slope: vertical line (x = constant).
Parallel lines: same slope (m₁ = m₂), different y-intercepts.
Perpendicular lines: slopes are negative reciprocals: m₁ × m₂ = −1. Slope 2/3 → perpendicular slope = −3/2.
Slope and quadrants: a line with positive slope must pass through Q1 and Q3. A line with negative slope must pass through Q2 and Q4. A line with positive y-intercept passes through Q1 and Q2; negative y-intercept through Q3 and Q4.

C) Equations of a Line

Slope-intercept: y = mx + b. m = slope, b = y-intercept (the value of y when x = 0).
Point-slope: y − y₁ = m(x − x₁). Use when you know one point and the slope.
Standard form: ax + by = c. Set y=0 to find x-intercept; set x=0 to find y-intercept.
To find the equation through two points: compute slope m = (y₂−y₁)/(x₂−x₁), then use point-slope with either point.
Example: through (1, 3) and (4, 9). Slope = (9−3)/(4−1) = 6/3 = 2. y − 3 = 2(x−1) → y = 2x + 1.

D) Distance and Midpoint

Distance between (x₁, y₁) and (x₂, y₂): d = √[(x₂−x₁)² + (y₂−y₁)²]. This is the Pythagorean theorem applied to coordinates.
Midpoint of segment from (x₁, y₁) to (x₂, y₂): M = ((x₁+x₂)/2, (y₁+y₂)/2). Average both coordinates.

E) Intersections

To find where two lines intersect, set their equations equal and solve the system. The solution (x, y) is the intersection point.

Parallel lines (same slope, different intercepts): no intersection.
Identical lines (same slope AND same intercept): infinitely many intersections.
Non-parallel lines: exactly one intersection.
Graphically: two equations with a unique solution represent two lines crossing at one point.

F) Parabolas

y = ax² + bx + c is a parabola. If a > 0, opens upward (U-shape, vertex is minimum). If a < 0, opens downward (∩-shape, vertex is maximum). X-intercepts: set y = 0 and solve for x. Y-intercept: set x = 0, giving y = c.

Example: y = x² − 4x + 3. Opens up (a=1>0). x-intercepts: (x−1)(x−3)=0 → x=1 and x=3. y-intercept: y=3. Vertex x = −(−4)/(2×1) = 2; y = 4−8+3 = −1 → vertex at (2, −1) (minimum).

G) Circles

A circle with center (h, k) and radius r has equation: (x − h)² + (y − k)² = r².

Standard form centered at origin: x² + y² = r².
To find the center and radius from a general equation, complete the square on both x and y terms.
Example: x² + y² − 6x + 4y = 3 → (x²−6x+9) + (y²+4y+4) = 3+9+4 → (x−3)² + (y+2)² = 16. Center (3, −2), radius 4.

H) Pythagorean Triples to Know

Right triangles with integer sides appear frequently. Memorize these and their multiples:

3-4-5 (and multiples: 6-8-10, 9-12-15, 5-12-13, 8-15-17)
5-12-13
7-24-25
8-15-17
Special right triangles: 45-45-90 has sides 1:1:√2; 30-60-90 has sides 1:√3:2.

In coordinate geometry, the distance formula IS the Pythagorean theorem — use triples to find distances without computing the square root when the values match.

Coordinate Geometry