Circle Notes

Introduction

In the study of coordinate geometry, the circle is a fundamental topic that frequently appears in the JEE Main Mathematics syllabus. This document will provide a comprehensive guide to understanding the properties, equations, and various forms of a circle. Each section will break down complex ideas into manageable parts, ensuring a thorough grasp of the topic.

Basic Definition and Equation of a Circle

A circle is defined as the set of all points in a plane that are equidistant from a given point called the center. The fixed distance from the center to any point on the circle is called the radius.

Standard Equation of a Circle

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

$$ (x - h)^2 + (y - k)^2 = r^2 $$

General Form of a Circle

The general form of the equation of a circle is:

$$ x^2 + y^2 + 2gx + 2fy + c = 0 $$

Here, the center and radius can be derived as follows:

• Center: $(-g, -f)$
• Radius: $\sqrt{g^2 + f^2 - c}$

Parametric Form of a Circle

The parametric form of the circle with center $(h, k)$ and radius $r$ is given by:

$$ x = h + r\cos\theta $$ $$ y = k + r\sin\theta $$

where $\theta$ is the parameter varying from $0$ to $2\pi$.

Tangents and Normals to a Circle

Equation of Tangent

The equation of the tangent to the circle $x^2 + y^2 = r^2$ at the point $(x_1, y_1)$ is given by:

$$ xx_1 + yy_1 = r^2 $$

For a circle with center $(h, k)$ and radius $r$, the tangent at $(x_1, y_1)$ is:

$$ (x - h)(x_1 - h) + (y - k)(y_1 - k) = r^2 $$

Equation of Normal

The normal to a circle at a point is the line perpendicular to the tangent at that point. For the circle $x^2 + y^2 = r^2$ at $(x_1, y_1)$, the normal is:

$$ \frac{x - x_1}{x_1} = \frac{y - y_1}{y_1} $$

Power of a Point

The power of a point $(x_1, y_1)$ with respect to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is given by:

$$ P = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c $$

If $P = 0$, the point lies on the circle. If $P > 0$, the point is outside the circle, and if $P

< 0, the point is inside the circle.

Chord of Contact

If two tangents are drawn from an external point $(x_1, y_1)$ to a circle, the line segment joining the points of tangency is called the chord of contact. The equation of the chord of contact for the circle $x^2 + y^2 = r^2$ is:

$$ xx_1 + yy_1 = r^2 $$

Important Properties and Theorems

Diameter and Circumference

• Diameter: $D = 2r$
• Circumference: $C = 2\pi r$

Area

The area of a circle with radius $r$ is given by:

$$ A = \pi r^2 $$

Arc Length and Sector Area

For an angle $\theta$ (in radians) subtended at the center, the arc length $L$ and the area of the sector $A_s$ are given by:

$$ L = r\theta $$ $$ A_s = \frac{1}{2}r^2\theta $$

Conclusion

Understanding the properties and equations of a circle is crucial for solving a variety of problems in coordinate geometry. Mastery of these concepts will provide a strong foundation for tackling JEE Main Mathematics questions on this topic. Practice is key, so work through multiple problems to reinforce these concepts.