**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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