Coordinate Geometry

What Is Coordinate Geometry

Coordinate geometry, also called analytical geometry, is the branch of mathematics that studies the position of points, lines, and curves on a plane using coordinates $(x, y)$. It connects algebra and geometry, making it easier to calculate distance between points, midpoints, slopes of lines, and equations of straight lines and circles in mathematics. Coordinate geometry is widely used in physics to represent motion and forces, in engineering and construction for designing structures, and in astronomy to determine distances and positions of celestial objects. It also helps in graphical representation of functions and shapes, giving a clear visual understanding of mathematical concepts.

Coordinate Geometry - Basic Terminologies and Definitions

Straight Lines: In coordinate geometry, a straight line is represented by an equation in the form $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept. For example, the line $y = 2x + 3$ cuts the y-axis at $3$ and rises 2 units for every 1 unit it moves along the x-axis.

Inclination of a Line: In a coordinate plane, a line forms an angle with the positive direction of the x-axis. This angle, measured anticlockwise, is called the inclination of the line. For example, if a line rises 3 units vertically for 4 units horizontally, its inclination $\theta$ can be found using $ \tan \theta = \frac{3}{4} $.

It forms the basis and foundation of advanced topics such as conic sections including hyperbola, parabola, circle and ellipse, that play a major role in higher level mathematics.

Circle: A circle is the set of all points in a coordinate plane that are equidistant from a fixed point called the center. Example: The circle with center at $(0,0)$ and radius $5$ has the equation $x^2 + y^2 = 25$.

Basics of the Cartesian Plane

The Cartesian plane is the foundation of coordinate geometry, allowing us to locate points, lines, and shapes using algebra and geometry together. It helps students visualize and analyze geometric problems easily.

X-axis, Y-axis, and Origin

The Cartesian plane has two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). The intersection point is the origin $(0,0)$, and every point is written as $(x, y)$.

Quadrants and Coordinates of a Point

The plane is divided into four quadrants:

• Quadrant I: $x > 0, y > 0$

• Quadrant II: $x < 0, y > 0$

• Quadrant III: $x < 0, y < 0$

• Quadrant IV: $x > 0, y < 0$
  Example: The point $(-3, 4)$ lies in Quadrant II.

Plotting Points on the Cartesian Plane

To plot a point $(x, y)$, start at the origin, move $x$ units along the x-axis, then $y$ units along the y-axis. This method helps in graphical visualization of points and lines.

Distance and Midpoint Formulas

Distance and midpoint formulas are essential to calculate lengths of segments and middle points, widely used in geometry, physics, and engineering.

Distance Between Two Points $(x_1, y_1)$ and $(x_2, y_2)$

The distance formula calculates the length of a line segment:
$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $
Example: Distance between $(1, 2)$ and $(4, 6)$ is $5$.

Midpoint of a Line Segment

The midpoint formula finds the point exactly in the middle of a segment:
$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $
Example: Midpoint of $(2, 3)$ and $(6, 7)$ is $(4, 5)$.

Slope of a Line and Angle with X-Axis

Slope tells us how steep a line is, and the angle of inclination helps in understanding the line’s direction on the plane.

Formula for Slope $(m = \frac{y_2 - y_1}{x_2 - x_1})$

Slope measures how steep a line is. Positive slope rises, negative slope falls.

Positive, Negative, Zero, and Undefined Slopes

• Positive slope: Line rises from left to right

• Negative slope: Line falls from left to right

• Zero slope: Horizontal line

• Undefined slope: Vertical line

Angle of Inclination of a Line

The angle $\theta$ a line makes with the positive x-axis is:
$ m = \tan \theta $

Equations of a Line

Equations of a line allow us to represent lines algebraically, making it easier to solve geometric problems and calculate intersections.

Slope-Intercept Form $(y = mx + c)$

Represents a line with slope $m$ and y-intercept $c$.

Point-Slope Form $(y - y_1 = m(x - x_1))$

Used when a line passes through a point $(x_1, y_1)$ with slope $m$.

General Form $(Ax + By + C = 0)$

A standard form for any straight line.

Horizontal and Vertical Lines

• Horizontal line: $y = k$

• Vertical line: $x = h$

Section Formula and Coordinate Geometry Applications

The section formula helps divide a line segment internally or externally and is widely used in physics, engineering, and real-life mapping problems.

Internal Division of a Line Segment

A point dividing a segment internally in the ratio $m:n$:
$ P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) $

External Division of a Line Segment

A point dividing a segment externally in the ratio $m:n$:
$ P = \left( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n} \right) $

Important Formulas in Coordinate Geometry

Equation of a Line: The general equation of a line is $ax + by + c = 0$.

Distance Formula: The distance $D$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is
$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Slope of a Line: For the line $ax + by + c = 0$, the slope $m$ is
$m = -\frac{a}{b}$.

Slope of parallel and perpendicular lines:

Angle Between Two Lines: If two lines have slopes $m_1$ and $m_2$, the angle $\theta$ between them is
$\theta = \tan^{-1} \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$.

Distance of a Point from a Line: The distance $d$ of a point $(x_1, y_1)$ from the line $ax + by + c = 0$ is
$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$.

Distance Between Two Parallel Lines: For two parallel lines of slope $m$, $y = mx + c_1$ and $y = mx + c_2$, the distance $d$ between them is
$d = \frac{|c_1 - c_2|}{\sqrt{1 + m^2}}$.

Recommended Books for Coordinate Geometry

Find the best books for Class 10 coordinate geometry that simplify concepts, provide solved examples, and enhance problem-solving skills for board and competitive exams.

NCERT Resources

Access NCERT textbooks, solutions, and practice exercises for coordinate geometry to build strong fundamentals and excel in Class 10 mathematics exams.

NCERT Maths Class 10 Notes for Chapter 7 - Coordinate Geometry

NCERT Maths Class 10 Solutions for Chapter 7 - Coordinate Geometry

NCERT Maths Class 10 Exemplar Solutions for Chapter 7 - Coordinate Geometry