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A circle is a two-dimensional geometric shape with all the points lying at the same distance from a fixed point. This fixed point is termed the "center" of the circle. The circle can also be stated as two foci meeting at the same point with the line segment bent in a circular shape. Let us understand the different properties of a circle in order to get an inclusive insight into the problems we are going to solve.
| Table of Content |
Keyterms: Circle, Geometric shape, Center, Radius, Rotational symmetry, Annulus, Sector, Segment, Chord
Definition of Circle
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It is essential to understand that whenever a line passes through a circle, a line of reflection is formed every time. Not only this, but it also has rotational symmetry at the center of every angle. Now let us talk about the general equation of a circle,
Circle
Let us take point C as the center of the circle.
The line between points C and P in the diagram is the radius which can be taken as r.
Therefore,
By applying the distance formula we get,
(x - h)2 + (y - k)2 = r 2
This is the standard equation of a circle with center C lying at coordinates (h, k).
Discover about the Chapter video:
Conic Sections Detailed Video Explanation:
Different Segments of Circle
Let us now discuss the different parts of the circle:
- Annulus: Annulus is a ring-shaped part. It is derived as the region between two concentric circles as shown in the figure below.
Annulus
- Arc: It is a term for the connected curve part of a circle.
- Sector: Sector is the region that is enclosed by two radii and an arc.
Sector
- Segment: This region of a circle is enclosed by an arc and a chord.
- Centre: The fixed point at the center of a circle.
- Chord: Chord is known as the line whose endpoints are situated on the circle.
- Radius: The line from the center of a circle to any point on the circumference is known as radius.
Radius
- Diameter: The longest chord of a circle is known as its diameter. It is a line segment having endpoints that lie on the circle and passes through the center.
- Secant: This line segments the circle into two halves and is also known as an extended chord.
- Tangent: It is known as the line that touches the circle at one point.
Properties of a Circle
Following are some certain properties of this 2D geometric shape known as a circle:
- The diameter of a circle segments the circle into two equal halves.
- The outer line or circumference of a circle is at an equal distance from the center.
- When two or more circles have different radii, they are similar to each other.
- However, circles that have equal radii measurements are termed as “congruent”.
- Diameter = 2 * radius and it is the longest chord of a circle.
Also Read:
| Related Concepts | ||
|---|---|---|
| Tangent Circle Formula | Circle Definition | Tangent to a Circle |
| Chord of Circle | Central Angle of a Circle | Great Circle Formula |
Important Formulas
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Following are the essential formulas that must be kept in mind while solving the problems of the circle:
| Particulars | Formula / Equation | Diagram |
|---|---|---|
| Diameter (d) | d = 2r (where r = radius of the circle) |
Diameter |
| Circumference (C) Def: We know about the perimeter of straight-lined shapes. The circumference is a circle’s perimeter. The distance that is around a circle is known as its circumference. | C = πd = 2 π r (where d = diameter and r = radius), and π = 3.14 or 22/7 |
Circumference |
| Area (A) Def: As we know area is the occupied space of any shape. Similarly, the area of a circle is the space occupied by the circle. | Area of a circle or A = πr2 (where r = radius if the circle) |
Area |
| If the circle passes through the origin. | The equation for such circle will be, x2 + y2 - 2hx - 2ky = 0 | |
| If the circle touches the X-axis at one point. | The equation for this circle will be, x2 + y2- 2hx - 2ay + h2=0 | |
| If the circle only touches the Y-axis at one point. | Then the equation for such circle will be, x2 + y2- 2ax - 2ky + k2 = 0 | |
| If the circle touches both the axis and the center lies at the same distance from both the X and Y-axis. | Then the equation for this kind of circle will be, x2 + y2 - 2ax - 2ay + a2=0 | |
| When the X-axis passes through the center of the circle segmenting the circle into two equal halves. | Then the equation for such circle will be, x2 + y2 - 2ax = 0. | |
| When the circle touches the origin (O) and the Y-axis divides the circle into halves by acting as the diameter. | Equation for the circle will be, x2 + y2 - 2ay = 0 | |
| If the circle cuts the origin and both the axis at respective points | Then the equation for such circle will be, x2 + y2 - by = 0 | |
| If the center of the circle is the origin of the two axes. | Then the equation will be, x2 + y2 = r2 |
Also Read:
Sample Questions
Ques: If the radius of a circle is 6 cm then find the area of that circle. (1 mark)
Ans: Radius or r = 6cm
Area or A = πr2
A = (3.14 * 62) cm2
A = 113.04 cm2
Ques: Find the diameter and circumference of a circle whose radius is 7 cm. (1 mark)
Ans: Diameter or d = 2r
= (2 * 7) cm
= 14 cm
Circumference or C = 2πr
= 2 * 22/7 * 7
= 44 cm.
Ques: If the radius of a circle is 4 and the center is (– 3, 4) then find the equation of the circle. (1 mark)
Ans: We know the standard equation of a circle which is,
(x - h) 2 + (y - k) 2=r 2
Here h = – 3, k = 4, and r = 4.
Therefore, the solution will be
(x + 3)2 + (y – 4)2 = (4)2 or 16
Ques: Find the equation of a circle whose center is (1, 1) and radius is 2 (2 marks)
Ans: We know,
(x - h)2 + (y - k)2=r 2
Here, h=1,
k = 1,
and r = 2
Therefore the solution for the circle’s equation will be,
(x - 1)2+(y - 1)2 =2
Ques: The equation of a circle is x2+y2+8x+10y-8=0 then find the radius and the centre of the circle. (2 marks)
Ans: From the mentioned equation we can say that,
x2 + y2+8x + 10y - 8=0
-
(x2+8x) + (y2+10y) = 8
Now further completing the squares in the equation we will get,
(x2 + 8x + 16) + (y2 + 10y + 25) = 25 + 8 + 16
=> (x+4)2 + (y+5)2 = 49
=> {x – (– 4)}2 + {y – (–5)}2 = 72
Therefore, comparing the above equation with the standard equation of the circle we can say that,
The Center of the circle is ( – 4, – 5) and the radius is 7.
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