COORDINATE GEOMETRY
Frequently Asked Questions
Coordinate Geometry (Analytical Geometry) is the branch of mathematics that represents points, lines, and shapes using numerical coordinates on a plane.
The Cartesian plane is a two-dimensional plane formed by two perpendicular number lines: the x-axis and the y-axis.
Coordinates are ordered pairs (x, y) that represent the position of a point on the Cartesian plane.
The x-axis is the horizontal axis on the coordinate plane.
The y-axis is the vertical axis on the coordinate plane.
The origin (0, 0) is the point where the x-axis and y-axis intersect.
Abscissa is the x-coordinate of a point.
Ordinate is the y-coordinate of a point.
The plane is divided into four quadrants numbered counterclockwise starting from the top-right region.
Quadrant I (+,+), Quadrant II (-,+), Quadrant III (-,-), Quadrant IV (+,-).
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
To find the distance between two points on the coordinate plane.
\( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \).
It finds the exact center between two given points.
For a point dividing line segment in ratio m:n internally: ( P = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) ).
When the dividing point lies outside the line segment; the formula uses (m-n) instead of (m+n).
\( \sqrt{x^2 + y^2} \).
When a point divides a line segment between the endpoints.
Use the midpoint formula.
Yes, depending on the quadrant where the point lies.
By checking the signs of x and y coordinates.
They represent the same point.
Used in GPS, navigation, mapping, architecture, engineering, computer graphics, and robotics.
Find distances between vertices using the distance formula and add them.
Yes—horizontal lines use (
A set of evenly spaced horizontal and vertical lines forming squares on the plane.
Compute distances AB, BC, AC. If AB + BC = AC, they are collinear.
In higher classes: slope = \(\frac{y_2 - y_1}{x_2 - x_1}\).
No, slope is part of Class 11, but helpful for conceptual understanding.
- Reflection in x-axis ? (x, -y) - Reflection in y-axis ? (-x, y) - Reflection in line y=x ? (y, x) - Translation ? (x+a, y+b).
Shifting a point by adding/subtracting values to x and y coordinates.
Flipping a point across an axis or line.
Not formally, but included here for conceptual completeness and competitive exam utility.
Formula used in higher classes: \( \frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|\)
Distance, midpoint, and section formula questions; identifying quadrants; plotting points.
Distance formula, as it appears most frequently in board exams.
Move horizontally to x-value, then vertically to y-value.
Order matters: (x, y) refers to horizontal then vertical movement.
A straight path connecting two points represented by endpoints’ coordinates.
Set of points satisfying a certain condition (used in higher geometry).
To eliminate negative values and apply the Pythagorean theorem.
The line segment between them is vertical.
The line segment is horizontal.
It visually represents equations and solutions.
It numerically explains geometric properties.
Use section formula with ratio 1:2.
Yes—this gives the endpoint.
It shows the point exactly equidistant from both endpoints.
To locate points, calculate distances, and build geometric models.
Cartesian or rectangular graph paper.
They provide exact positions for vertices of polygons.
Maps use latitude–longitude grids, similar to coordinate grids.
Proving triangles are isosceles/equilateral.
Verify Pythagoras theorem using three distances.
Use the midpoint formula.
Use special Pythagorean triplets like (3,4,5), (6,8,10).
Sign mistakes, reversing (x, y), incorrect substitution in formulas.
Always mark the quadrant first before solving.
GPS, Google Maps, robotics, drones, animations, video games.
To analyze geometrical objects using algebraic formulas.
No, distance is always non-negative.
A pair (x, y) where order matters: x first, then y.
Formulas are direct and questions require simple substitution.
Distance formula, midpoint formula, section formula, quadrant rules.
Re-check signs and compare with diagram or rough sketch.
Usually 2–3 questions (3–5 marks combined).
Not compulsory, but helpful for reducing mistakes.
Logical reasoning, visualization, and analytical skills.
Translation: (x+a, y+b).
Center point of a line segment that splits it into two equal parts.
To determine how far a point lies between two endpoints.
Yes, but extended for three coordinates (x, y, z).
Ensure the dividing point lies between endpoints for internal division.
It should lie between the endpoints in both x and y values.
Motion, vectors, and forces are represented on coordinate planes.
