JEE Maths Notes: Chapter on Circle System

1. Introduction to Circle System

The Circle System in mathematics deals with the properties, equations, and theorems related to circles. Circles are fundamental geometric shapes, and their study plays a crucial role in coordinate geometry, algebra, and calculus. A circle is defined as the set of all points equidistant from a fixed point known as the center. Understanding the equation of a circle and its various forms is important for solving problems involving tangents, chords, and intersections with other curves.

2. Equation of a Circle

The general equation of a circle in the Cartesian coordinate system is given by:

x² + y² + 2gx + 2fy + c = 0

Here, g, f, and c are constants, and the center of the circle is at the point (−g, −f). The radius r of the circle can be calculated as:

r = √(g² + f² - c)

This equation can be rewritten in various forms, including the standard form and the general form, depending on the values of g, f, and c.

3. Standard Equation of a Circle

The standard equation of a circle with center (h, k) and radius r is given by:

(x - h)² + (y - k)² = r²

This equation is particularly useful for identifying the center and radius of a circle. It is derived from the distance formula and is frequently used in problems involving geometric loci and distance calculations.

4. General Equation of a Circle

The general equation of a circle, when expanded, is:

x² + y² + 2gx + 2fy + c = 0

From this equation, the center and radius can be found using the formulas:

• Center: (−g, −f)
• Radius: r = √(g² + f² - c)

This form is widely used to derive the properties of the circle, such as tangents, normal lines, and intersections with other curves.

5. Properties of a Circle

Some important properties of a circle include:

• Center: The fixed point that is equidistant from all points on the circle.
• Radius: The distance from the center to any point on the circle.
• Chord: A line segment joining two points on the circle. The longest chord is the diameter.
• Diameter: A special chord passing through the center of the circle. It is twice the length of the radius.
• Tangent: A line that touches the circle at exactly one point.
• Secant: A line that intersects the circle at two distinct points.

6. Tangents to a Circle

A tangent to a circle is a straight line that touches the circle at exactly one point. The key properties of tangents include:

• The tangent at any point on the circle is perpendicular to the radius drawn to the point of contact.
• The length of the tangent from an external point to the point of tangency can be calculated using the formula:

Length of tangent = √(OP² - r²)

Where OP is the distance from the external point to the center, and r is the radius of the circle.

7. Length of Chord

The length of a chord in a circle can be determined using the perpendicular distance from the center to the chord. The formula for the length of a chord at a perpendicular distance d from the center is:

Length of chord = 2√(r² - d²)

Where r is the radius of the circle and d is the perpendicular distance from the center to the chord.

8. Power of a Point

The power of a point with respect to a circle is defined as the square of the distance from the point to the center of the circle minus the square of the radius. It is given by:

Power of point P = OP² - r²

Where OP is the distance from the point P to the center, and r is the radius of the circle. This concept is useful in determining the position of a point relative to a circle and in solving problems involving tangents and secants.

9. Equation of a Circle in Polar Coordinates

In polar coordinates, the equation of a circle with center at (r₀, θ₀) and radius R is given by:

r² - 2r₀r cos(θ - θ₀) + r₀² = R²

This form is useful for analyzing circles in polar coordinate systems, which is commonly used in problems involving rotation, oscillations, and circular motion.

10. Circle and Its Intersection with Other Curves

The intersection of a circle with other curves such as straight lines, parabolas, and ellipses is an essential topic in geometry. The points of intersection can be found by solving the equations of the circle and the other curve simultaneously. The nature of the intersection (whether it results in two points, one point, or no solution) depends on the relative positions and equations of the curves.

11. Applications of Circle System

The concept of the Circle System has several practical applications, including:

• Navigation: Using circular motion to determine locations and bearings.
• Engineering: Designing gears, pulleys, and circular components in machinery.
• Physics: Modeling circular motion, rotation, and oscillations.
• Astronomy: Analyzing the orbits of planets and moons, which are elliptical but can be approximated as circular in many cases.

12. Practice Problems and Examples

This section provides a series of practice problems to help reinforce understanding of the Circle System. Problems include finding the equation of a circle, determining the length of tangents, solving for intersections with other curves, and using properties like the power of a point and the length of a chord. Step-by-step solutions are provided for each example.

13. Summary and Key Takeaways

In summary, the Circle System is an important topic in coordinate geometry, focusing on the properties and equations related to circles. Key concepts include the equation of a circle, tangents, chords, secants, and the power of a point. Understanding the geometric properties of circles and their relationships with other curves is essential for solving complex problems in mathematics and for applications in various fields.