# Chapter 9 – Circles

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

You might have come across this shape whose all points are inscribed in a plan in such a manner that the distance from a given point to the centre is in curvature. Majority of surroundings have a lot of circular shapes; so we can observe this in our day-to-day phenomenon that the distance is always constant from the given point from the centre to the trace of curvature. This shape is known as a circle, and it can be seen in wheels, rings, buttons and pies. In other words, a circle is a shape whose all points lie at an equal distance from a centre point. The line segment that has endpoints the Circle’s boundary and passes through the midpoint is called diameter; whereas, the line segment from the centre point to any point of a circle is known as radius. Let us explore more about circles in this article.

#### Equation of Circle

Standard equation of a circle with the centre at (a,b) and radius r can be stated as (x-a)2 + (y-b)2 = r2 . If the centre point is noted at point O = (a,b)  in a plane, construction of a unique Circle with radius r is easy. Check out videos on MSVgo to understand the concept behind this.

Parametric Equation of Circle

Parametric equation of the circle is derived from finding the coordinates of any point of Circle and relating it with the coordinates of the parameter of the circle. Considering the following Circle with centre at O(0,0) and radius=r. Let P(x,y) be any point on the parameter of the circle in such a way that OP makes the angle of θ at the X-axis. With the application of trigonometry, x=rcos θ and y = rsin θ. Check out videos on MSVgo for better understanding of the Parametric Equation of Circle.

#### Equation of Tangent of Circle

A tangent is a perpendicular line to the radius that touches at coordinates of the parameter of the circle. Being a straight, the equation for the tangent of the circle is y= mx+c. If P (x,y) is the point where a tangent touches the Circle, m can be derived using the perpendicular gradients and value of c can be derived by solving the equation.

Equation of Circle Under Different Conditions

Following is the list of the equations for Circle under different conditions:

ConditionsEquation
If both axes are at the centre of a, a and the radius is also a(x-a)2 + (y-b)2 = a2
If x-axis is at the centre of the Circle with α, a being the centre with a as radius(x-α)2 + (y-b)2 = a2
If the Circle touches x-axis with the centre at α, β with radius a(x-a)2 + (y- β)2 = a2

#### Equation of Tangents and Normal

The normal line equation to the curve at some particular point can be coined as the line that goes through a point of parameter and is perpendicular to the tangent. The equations of the tangents and normal for a Circle is x2+y2+2gx+2fy+c=0. For more information on this topic, browse videos on MSVgo.

#### Radical Axis to the Two Circles

The radical axis for two Circles can be detailed as the line by which the stated circles are connected perpendicularly. On the other hand, the radical axis for two disjoint circles can be defined as the locus of points that draw tangents on both the circles with equal lengths. The equation of radical axis of two Circles is 2x(g1 -g2) + 2y (f1 – f2) + c1 – c2 = 0 with a straight line at S1 -S2 = 0. Check out videos on MSVgo for better understanding of the concept.

#### Summary

A circle can be defined as the collection of all points in a plane which are at a constant distance from a fixed point which is at the centre of this shape. The constant distance referred is known as the radius and the line segment joining to the opposite side is known as diameter. The line that meets Circle only at one point is known as a tangent. The radical axis for two disjoint circles is the locus of the points of tangent that touch both the circles at equal lengths. In simple words, circles are enclosed curves who have equidistant from a fixed centre. The circle topic is very easy to understand once you grasp the basic concepts; we hope this article helps you understand the fundamentals of the topic.

#### FAQs

• What are the Circles in math?

A circle is a mathematical shape whose all points are in the same plane and lie at equal distance from its centre point. The core reason for the same is that the Circle comprises points only at the border in a circular pattern.

• What are the formulas for Circles?

Circumference of Circle: 2x π x r in which π  is pi and r is the radius

Area of Circle: π × r2 in which π  is pi and r is the radius

• What is special about Circles?

The Circle is a shape with the largest area being the perimeter regarding isoperimetric inequality. A circle is also special because it is highly symmetric as its rotational symmetry is present at the center from every angle.

• How are Circles used in real life?

Ferris Wheel is one of the most common examples of Circle at its outer rim has equidistant center similar to Circle. Circles are seen as the divine shape with the natural balance due to their symmetry as well as because of their significance in structure building throughout history.

• What are the 8 Circle theorems?

The 8 Circle theorems are:

1. Angles are at the centre or circumference of the Circle
2. Angles are in semiCircle
3. Angles are in the same segment
4. Angles are in cyclic quad-lateral.
5. Lengths of the tangents
6. Angles between tangents of the Circles and radius.
7. Alternate segment Theorem
8. Perpendicular point from the centre tends to bisect the chord.

Learning from textbooks and notes gets very reductive and a tedious task. For understanding and grasping concepts, students require new mediums to keep the educational process interesting and fun. This is where MSVgo helps you. MSVgo has a library full of videos for your syllabus; the explanatory videos are a new and fun way of learning. For more information, the MSVgo app or the official website.

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