# Formula Sheets for General Aptitude (Part A): Coordinate Geometry , Download PDF!

By Astha Singh|Updated : March 10th, 2023

General Aptitude Formula Sheets: During the preparation, the candidates study different formulas to solve problems, but at the last moment, these formulas might not be remembered by the candidates due to exam fear or pressure. We at BYJU'S Exam Prep do not want our students to lag anywhere during the preparation, so we have come up with a concept of a Formula Sheet that will help them revise the important formulas at the last moment. This formula sheet will be a short revision tool and contain only important formulas that need to be studied at the last minute to boost the score. Our experienced subject-matter experts have meticulously designed this CSIR NET General Aptitude Formula Sheet to provide you with the best authentic material.

In this article, we will cover the CSIR NET General Aptitude Most Important Formulas of Coordinate Geometry. Aspiring candidates can check all the most important formulas of the Coordinate Geometry for the last-minute revision. Scroll down the full article to find out!

## Formula Sheet On Coordinate Geometry

Coordinate Plane: A coordinate plane is a 2-D plane formed by the intersection of a vertical line called the y-axis and a horizontal line called the x-axis. These are perpendicular lines that intersect each other at zero, and this point is called the origin O (0, 0). The axes cut the coordinate plane into four equal sections, and each section is known as quadrant

The two-dimensional plane is called the Cartesian plane, or the coordinate plane and the
axes are called the coordinate axes or x-axis and y-axis. The given plane has four equal divisions by origin called quadrants.
• The horizontal line towards the right of the origin (denoted by O) is a positive x-axis.
• The horizontal line towards the left of the origin is the negative x-axis.
• The vertical line above the origin is a positive y-axis.
• The vertical line below the origin is a negative y-axis.
• The x-coordinate or abscissa of a point is its perpendicular distance from the y-axis
measured along the x-axis.
• The Y-coordinate or ordinate of a point is its perpendicular distance from the x-axis
measured along the y-axis.
• In stating the coordinates of a point in the coordinate plane, the x-coordinate comes first,
and then comes the Y-coordinate. We place the coordinates in brackets as (x, y). Distance between two points (x1, y1), (x2, y2):

Section Formula: The co-ordinates of a point P(x ,y), dividing the line segment joining
the two points A(x1, y1) and B (x2, y2) internally in the ratio m : n are given by

The co-ordinate of the point P(x, y), dividing the line segment joining the two points
A(x1, y1) and B (x2, y2) externally in the ratio m : n are given by

Trisection Formula of a line segment: If points P and Q lie on line segment AB
divide it into three equal parts that means, if AP = PQ = QB then the points P and Q are
called Points of Trisection of AB

Here, P divides AB in the ratio of 2: 1 and Q divides AB in the ratio of 1: 2. Now use the section formula for finding the coordinates of P and Q.
Reflection of a point in the axes and origin:
1. Reflection in the X-axis: Here, the x-axis represents the plain mirror. When point M is reflected in the x-axis, the image M’ is formed in the horizontally opposite quadrant whose coordinates are (h, -k). Thus, when a point is reflected in the x-axis, then the x-coordinate remains the same, but the Y co-ordinate becomes negative

Thus, the image of point M (h, k) is M'(h, -k). Rule:
(i) Retain the abscissa i.e., x-coordinate.
(ii) Change the sign of ordinate i.e., y-coordinate.
2. Reflection in the Y-axis: Here, the y-axis represents the plane mirror. when point M is reflected in the y-axis, the image M' is formed in the vertically opposite quadrant whose coordinates are (-h, k). Thus, when a point is reflected in the y-axis, then the y-coordinate remains the same and then the x-coordinate become negative.

Thus, the image of M (h, k) is M'(-h, k).
Rule:
(i) Change the sign of abscissa i.e., x-coordinate.
(ii) Retain the ordinate i.e., y-coordinate.
3. Reflection through Origin: When a point is reflected in origin, both x-coordinate and y-coordinate change. Thus, the reflection of M (h, k) is M’ (-h, -k) in the origin.

Rule:
(i) Change the sign of abscissa i.e., x-coordinate.
(ii) Change the sign of ordinate i.e., y-coordinate.
The coordinates of the midpoint of the line formed by A(x1, y1), B(x2, y2): Here, the P point divides the line segment AB into ratio 1:1. Thus, m = n = 1.

Area of triangle whose coordinates are A(x1, y1), B(x2, y2), C(x3, y3):

Collinear points: Three or more points that lie on the same straight line are called collinear points. There are two methods to find if the three points are collinear:
(i) Slope formula method: Three or more points are collinear if the slope of any two pairs
of points is the same. Let three points be A, B and C, three pairs of points can be formed as
AB, BC and AC.
If the slope of AB = slope of BC = slope of AC, then A, B and C are collinear points. (ii) Area of triangle method: Three points are collinear if the value of area of the triangle formed by the three points is zero.
The slope of a line: If a line joins two points A(x1, y1) and B (x2, y2) then the slope of the
the line joining the two points.

The angle between two lines: If two lines have slopes m1 and m2 then the angle between
the two lines are given by

Note: If lines are parallel to each other then tanθ = 0°
If lines are perpendicular to each other then cotθ = 0°
Equation of line parallel to y-axis: The equation of a straight line to the x-axis and at a distance a from it, is given by X = a.

Equation of line parallel to x-axis: The equation of a straight line parallel to the y- axis and at a distance a from is given by Y = b.

Different types of Equations of line:
1. Normal equation of the line: ax + by + c = 0Note: Area of the triangle formed by co-ordinate axes and the line ax + by + c = 0 is given by .
2. Polar Form of an equation:

Co-ordinates of points in Polar Form: (𝐫𝐒𝐢𝐧 𝛉, 𝐫𝐂𝐨𝐬 𝛉)
3. Slope – Intercept Form: y = mx + c Where, m = slope of the line & c = intercept on Y-axis

4. Intercept Form:

5. Trigonometric form of equation of line, ax + by + c = 0

6. Equation of line passing through the point (x1, y1) & has a slope “m”: 𝐲 − 𝐲𝟏 = 𝐦(𝐱 − 𝐱𝟏)

7. Equation of two lines parallel to each other: Here, ax + by + c1 = 0 and ax + by + c2 = 0 represent the equations of two lines parallel to each other. “d” represents the distance between the two parallel lines.

Note: Here, coefficient of x & y will be same
8. Equation of two lines perpendicular to each other:
ax + by + c1 = 0
bx - ay + c2 = 0
Note: Here, the coefficient of x & y are opposite & in one equation there is a negative sign. Note: If m1, m2 are slopes of two perpendicular lines then m1.m2 = -1. The Distance of a Point from a Line: The length of the perpendicular from a point A(x1, y1) to a line with the equation ax + by + c = 0 is:

The Distance between two parallel lines: When two parallel straight lines with
equations ax + by + c1 = 0 and ax + by + c2 = 0, then the distance between them is given by:

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