Class 9 Mathematics – Circles & Coordinate Geometry Notes (Conceptual)

1. Circles

Definition

A circle is a set of all points in a plane that are at a fixed distance from a fixed point called the center.

• The fixed distance is called the radius.

Parts of a Circle

• Center: The fixed point from which all points on the circle are equidistant.

• Radius: Distance from the center to any point on the circle.

• Diameter: The longest distance across the circle passing through the center.

• Chord: A line segment joining any two points on the circle.

• Tangent: A line that touches the circle at exactly one point.

• Secant: A line that intersects the circle at two points.

• Arc: A part of the circumference of the circle.

• Sector: The region enclosed by two radii and the corresponding arc.

• Segment: The region enclosed by a chord and the corresponding arc.

Properties

• All radii of a circle are equal.

• The diameter is always twice the radius.

• A tangent is always perpendicular to the radius at the point of contact.

• The longest chord of a circle is the diameter.

Special Terms

• Circumference: The distance around the circle.

• Interior: The region inside the circle.

• Exterior: The region outside the circle.

2. Coordinate Geometry

Definition

Coordinate geometry is a branch of mathematics that uses a coordinate system to locate points in a plane and study geometric shapes.

Coordinate Plane

• A flat surface divided into four regions called quadrants by:

  • Horizontal line → X-axis

  • Vertical line → Y-axis

• The point where X-axis and Y-axis meet is called the origin.

Coordinates of a Point

• A point in the plane is represented by a pair of numbers (x, y):

  • x-coordinate: Position along the X-axis

  • y-coordinate: Position along the Y-axis

Distance & Midpoint Concept

• Distance: How far two points are from each other.

• Midpoint: The point exactly in the middle of two points.

Plotting Points

• Steps to plot a point:

  1. Start from the origin.

  2. Move along X-axis according to the x-coordinate.

  3. Move along Y-axis according to the y-coordinate.

  4. Mark the point.

Quadrants

• 1st Quadrant: (+, +)

• 2nd Quadrant: (−, +)

• 3rd Quadrant: (−, −)

• 4th Quadrant: (+, −)

Tips for Remembering

• Chord vs Diameter: All diameters are chords, but not all chords are diameters.

• Tangent property: Think of it as a “just touching” line”.

• Coordinates: X comes before Y, just like reading.