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Circles in Maths


Analytical Maths is basically the part of Algebra, it generally deals with the points, circles, lines, etc. The other name of analytical Geometry is coordinate geometry. Here we will study about the circles in maths. To understand the Circle in maths first we must go through its definition and its various parameters.
Chord- A Chord is a straight line which passes through the circle.
Radius- A Line Segment from a Center of Circle to its circumference is called as radius.
Diameter- Twice of radius is known as Diameter.

A circle can be defined as a line forming a closed loop and each Point on circumference is at equal distance from the center. The coordinates of center of the circle are (0, 0).
General form of Circles in maths: x2 + y2 + 2Dx + 2Ey + F = 0.
Now let us understand the other aspects of circle maths:
The perimeter of the circle is its circumference.
Circumference also has an arc.
Half of the circle is known as semi circle.
Intercepts: In standard form the circle intercepts the x-axis at (r, 0) and (-r, 0). In the standard form the circle intercepts the y- axis at (0, r) and (0, -r).
Eccentricity of circle: Let 'e' be the eccentricity of the circles, therefore
e = -- = 0,
Equation of a Tangent: Let us suppose the point on the circle A(x1, y1), then the equation form will be (x1 – a)(x – a) + (y1 - b)(y - b) = r2 .
If we need to find out the distance and angle in Analytical Geometry the below mentioned formula can be used:
d = √(r2 - r1)2 + (s2 - ­s1)2. Where‘d’ is the distance and r2, r1, s2, s1 are the coordinates of the plane of the geometry. This is all about circles maths.

Equation of Circle

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In mathematics, a round shape in which all the points on the boundary are at same distance from the center is defined as a Circle. The general equation of a circle: X 2 + Y 2 = r2, where ‘r’ represents the radius of a circle.
In Geometry, the unlike parabolas and circles always have X 2 and Y 2 terms.
It is necessary to remember that a circle is a locus of points. A circle in which all of the points are at fixed distance from the center is known as the radius of a circle.
The general equation for circle is given as:
⇒ (x – p)2 + (y – q)2 = r2;
Now put the value of p, q, r is 2, 3, 4 respectively then we get:
⇒ (x – 2)2 + (y – 3)2 = (4)2;
On further solving the equation of a circle then we get:
⇒ x2 + 4 – 4x + y2 + 9 – 6y = 16;
So the equation of a circle is:
⇒ x2 + y2 – 4x – 6y + 13 = 16;
We can also write it as:
⇒ x2 + y2 – 4x – 6y – 3 = 0;
This is the required equation of a circle.
We can also write the general form of a circle by putting the constant value in place of Numbers. So the equation of circle using constant is:
⇒ x2 + y2 + Ax + By + C = 0
Now we will see the equation of a Unit Circle:
As we know the radius of unit circle is always ‘1’, if the radius of a unit circle is more or less then ‘1’ then the circle is not unit circle.
The general equation of a unit circle is given as:
⇒ x2 + y2 = 1;
This is all about the equations of a circle.

Radius of Circle

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In mathematics, a round shape in which all the points on the boundary is equidistance form the center is known as circle. Now, we will see the Area of a Circle. The formula to find the area of circle is given as:
Area of a circle = ⊼r2, where ‘r’ is the radius of circle, and the value of π is 3.14.
Let’s check how to find the Diameter of a circle using the area of circle. The formula for finding the area of circle using the diameter is given as:
Area = ⊼ D2, where‘d’ is the diameter of a circle.
Suppose we have value of circumference, then we can find area of circle using circumference, which is given as:
Area =C2/4⊼
Where ‘c’ is the circumference of a circle;
Let’s see how to find the radius of a circle.
To find radius of a circle we need to follow some steps which are shown below:
Step1: To find the Radius of a Circle first of all it is necessary to find the area of a circle and value of ⊼.
Step2: When value of area of circle and pi is given then we can easily find the radius of circle.
Area = ⊼r2;
In the given formula, if we have value of area then we can easily find the radius of a circle.
R2 = area / ⊼;
R = √ area / ⊼;
When we put the value of area and pi we easily get the value of radius of a circle. We can also find the radius with help of circumference of a circle. Let’s see how we can determine the radius when circumference is given. Suppose we have circumference of a circle as 20, then radius will be calculated as:
Circumference of a circle = 2 ⊼ R;
R = circumference,
2 * ⊼
Now putting the value and we get:
R = 20/(2*3.14),
R = 20 / 6.28
R = 3.18 inch,
So radius of a circle is 3.18 inch.

Equation of Tangent Line to Circle

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Derivative of a function is defined as the change in rate of a function. The way of finding the derivative is termed as differentiation and differentiation is opposite process of Integration. Integration refers to collection of different identities into a single identity and its reverse is termed as differentiation, which defines the distribution of a single identity into sub identities or different identities. Differentiation of any term involves the derivation of an equation with respect to any particular variable.
Finding the equations like equation of normal, equation of Tangent, and Slope of the equation are among application of Derivatives. Slope of any line is defined as the inclination of any line. It comes under two dimensional Geometry as it is consists of x axis and y axis only.
The Slope in differential term is defined as the Ratio of change in ‘y’ with respect to change in x. It is represented by Δ y/ Δ x or d y/ d x. where,
Δ y = y2 - y1
Δ x = x2 – x1
Slope of any equation is represented by 'm' and 'tan θ' and the value of it is determined by Δ y/ Δ x.
Tangent is defined as the Straight Line at a Circle which does not passes through the circle but it touches the circumference of the circle at a particular Point. The equation of tangent line to circle is determined using the concept of slope. Its equation is given as:
y - y1 = (d y / d x) x – x1
Where, ‘y’ and ‘x’ are coordinate variables and (x1 , y1) is the point at the circle at which the tangent occurs.
The normal is defined as the line perpendicular to the tangent at the circle and its equation is represented as
y - y1 = (d x / d y) x – x1.

Center of Circle

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Circle is defined as the path traced by any Point at a constant distance in a curvature form i.e the distance of every point over that path remains same from the point in respect of which the curve is formed. A Circle divides a plane into two regions. Inner part is known as the interior of circle and outer part is known as the exterior of circle.
Concentric circles are also a part of circle family. Concentric circle refers to the circle into a circle. Number of circles can be embedded into a circle. The circle which lies in the center is known as center circle. And its radius will be minimum among all the radius.
Distance of the circle from the center is known as Radius of the circle. The distance of the radius from each point on the circle is constant. It is represented by ‘r’. The Diameter of the circle passes through the center and it is twice the radius of the circle which is represented by ‘d’ or ‘2r’. The boundary of the circle is known as the circumference of the circle and any line segment whose end points lies on the circumference of the circle is known as the Chord of the circle. The circumference of the circle is represented as:
Here, C is the circumference of the circle, r is the radius and π is a special variable whose value is 3.14 or 22/7.
And area of circle is represented by:
If we need to find the radius of the circle, whose circumference is given as 44 units then w can proceed as follows:
We know that C=2πr therefore, 44=2*(22/7)*r which gives (44*7)/44=r,
Thus, r=7 is the radius of the circle.
The circle may lie on origin as well as out of origin. Thus the equation of the circle whose center is the origin that is (0, 0) is represented by:
X2+y2= r2
And the Equation of Circle at any point say (c, d) is represented by:
(x-c)2 + (y-d)2 =r2.