Equation Of Circle General Form
The standard equation of a circle is given by:
(x-h) 2 + (y-yard) ii = r two
Where (h,k) is the coordinates of middle of the circumvolve and r is the radius.
Before deriving the equation of a circle, let us focus on what is a circle? A circle is a set of all points which are every bit spaced from a stock-still point in a airplane. The fixed point is called the center of the circle. The altitude between the center and whatsoever point on the circumference is called the radius of the circle. In this article, we are going to discuss what is an equation of a circle formula in standard form, and detect the equation of a circle when the centre is the origin and the heart is not an origin with examples.
- Equation of Circle
- Centre at Origin
- Centre not at Origin
- Full general Form
- Polar Equation
- How to Find the Equation?
- Solved Examples
- Practise Questions
- FAQs
What is the Equation of a Circumvolve?
A circumvolve is a closed curve that is drawn from the stock-still point chosen the heart, in which all the points on the curve are having the same distance from the centre point of the center. The equation of a circle with (h, k) centre and r radius is given by:
(ten-h) ii + (y-k) 2 = r 2
This is the standard course of the equation. Thus, if nosotros know the coordinates of the middle of the circle and its radius equally well, we tin hands find its equation.
Example: Say betoken (1,ii) is the eye of the circle and radius is equal to 4 cm. And then the equation of this circle will be:
(x-1) ii +(y-2) 2 = 4 2
(x 2 −2x+1)+(y 2 −4y+four) =16
X 2 +y 2 −2x−4y-eleven = 0
Function or Not
We know that there is a question that arises in example of circle whether being a function or not. It is clear that a circumvolve is non a function. Considering, a office is defined by each value in the domain is exactly associated with 1 signal in the codomain, but a line that passes through the circle, intersects the line at 2 points on the surface.
The mathematical way to describe the circle is an equation. Here, the equation of the circumvolve is provided in all the forms such as general form, standard course along with examples.
Equation of a Circle When the Centre is Origin
Consider an capricious point P ( x , y ) on the circle. Let ' a' be the radius of the circumvolve which is equal to O P .
We know that the distance between the point ( ten , y ) and origin ( 0 , 0 ) can be found using the distance formula which is equal to-
√[ x2 + y 2 ]= a
Therefore, the equation of a circle, with the center every bit the origin is,
x2 + y 2 = a two
Where "a" is the radius of the circle.
Alternative Method
Let us derive in another style. Suppose (ten,y) is a signal on a circle, and the center of the circle is at origin (0,0). Now if nosotros draw a perpendicular from bespeak (x,y) to the ten-centrality, then we go a correct triangle, where radius of the circle is the hypotenuse. The base of the triangle is the distance along ten-centrality and height is the distance forth the y-axis. Thus, past applying the Pythagoras theorem here, nosotros get:
x ii +y 2 = radius ii
Equation of a Circle When the Centre is non an Origin
Let C ( h , yard ) be the centre of the circumvolve and P ( x , y ) be any point on the circle.
Therefore, the radius of a circle is CP.
By using distance formula,
(ten-h) 2 + (y-k) two = CP 2
Let radius be 'a'.
Therefore, the equation of the circle with eye (h, yard)and the radius 'a' is,
(ten-h) 2 +(y-chiliad) 2 = a two
which is called the standard form for the equation of a circle.
Full general grade of Equation of a Circumvolve
The general equation of any type of circle is represented by:
x 2 + y two + 2 g x + two f y + c = 0 , for all values of g , f and c .
Adding g ii + f 2 on both sides of the equation gives,
x 2 + 2 m 10 + g 2 + y 2 + 2 f y + f two = g ii + f ii − c ………………(1)
Since, ( ten + m ) 2 = 10 2 + ii g x + 1000 2 and ( y + f ) 2 = y 2 + 2 f y + f two substituting the values in equation (ane), nosotros have
( x + g ) 2 + ( y + f ) 2 = g 2 + f two − c …………….(ii)
Comparing (two) with ( x − h ) 2 + ( y − k ) 2 = a two , where ( h , g ) is the center and ' a' is the radius of the circle.
h = − one thousand , k = − f
a 2 = grand 2 + f ii − c
Therefore,
x 2 + y 2 + 2 one thousand ten + 2 f y + c = 0 , represents the circle with heart ( − g , − f ) and radius equal to a 2 = g 2 + f two − c .
- If g two + f ii > c , and then the radius of the circle is existent.
- If g 2 + f 2 = c , then the radius of the circle is zilch which tells usa that the circle is a betoken that coincides with the eye. Such a type of circumvolve is chosen a point circumvolve
- g 2 + f 2 < c , then the radius of the circle become imaginary. Therefore, it is a circle having a real center and imaginary radius.
Polar Equation of a Circumvolve
To find the polar course of equation of a circle, supercede the value of x = r cos θ and y = r sin θ, in x2 + y2 = a2.
Hence, nosotros get;
(r cos θ)2 + (r sin θ)2 = aii
rtwo costwo θ + r2 sin2 θ = aii
r2 (costwo θ + sin2 θ) = aii
r2 (1) = a2 [Using trigonometry identity]
r = a
is the polar equation of a circle with radius a and center at the origin (0,0).
Video Lessons on Circles
Introduction to Circles
Parts of a Circle
Expanse of a Circle
All about Circles
Other Circle Formulas
Hither are some formulas are given for circle in terms of radius.
| Diameter | 2 x radius |
| Circumference | 2π (radius) |
| Surface area | π(radius) 2 |
How to Find the Equation of the Circumvolve?
To notice the equation of a circle given the radius and middle of the circle, we tin direct put the values in the standard form of the equation.
(x-h) 2 + (y-k) two = r two
Here, some solved problems are given to find the equation of a circle in both cases such as when the eye of a circle is origin and center is not an origin is given below.
Solved Examples
Example 1:
Consider a circle whose center is at the origin and radius is equal to 8 units.
Solution:
Given: Centre is (0, 0), radius is viii units.
We know that the equation of a circumvolve when the center is origin:
x2 + y 2 = a 2
For the given status, the equation of a circle is given equally
x2 + y 2 = eight 2
x2 + y 2 = 64, which is the equation of a circle
Example ii:
Find the equation of the circumvolve whose center is (3,5) and the radius is iv units.
Solution:
Here, the center of the circle is not an origin.
Therefore, the general equation of the circle is,
(x-iii) 2 + (y-five) 2 = 4 2
x 2 – 6x + nine + y 2 -10y +25 = 16
10 two +y 2 -6x -10y + 18 =0
Instance 3:
Equation of a circle is x 2 + y two − 12 x − 16 y + nineteen = 0 . Find the center and radius of the circumvolve.
Solution:
Given equation is of the form x 2 + y 2 + 2 g 10 + two f y + c = 0 ,
2 chiliad = − 12 , 2 f = − xvi , c = 19
k = − 6 , f = − 8
Center of the circumvolve is ( 6 , 8 )
Radius of the circle = √[ ( − 6 ) ii + ( − 8 ) 2 − nineteen ]= √[ 100 − nineteen] =
= √81 = 9 units.
Therefore, the radius of the circumvolve is 9 units.
Practice Questions on Equation of Circle
- Find the equation of a circle of radius 5 units, whose centre lies on the ten-axis and which passes through the bespeak (2, three).
- Notice the equation of a circumvolve with the centre (h, chiliad) and touching the x-axis.
- Testify that the equation xii + y2 – 6x + 4y – 36 = 0 represents a circumvolve. Also, find the centre and radius of the circumvolve.
To know more than almost circles download BYJU'S – The Learning App to learn with ease.
Frequently Asked Questions on Equation of a Circle
What is the equation for a circumvolve?
The equation for a circle is given past:
(10-h)two+(y-yard)2 = atwo
Where (h,1000) is the center and a is the radius of the circle.
What are the formulas for circles?
The circumference of a circumvolve is equal to ii (pi) of radius or pi of diameter.
The expanse of a circumvolve is equal to the pi of radius-squared.
What is the equation of a circle when the heart is at the origin?
At origin, the value of coordinates is (0,0), therefore, the equation of circle becomes:
(x-0)ii + (y-0)2 = rtwo
x2 + y2 = r2
If (x-iv)two+(y+7)two=nine is the equation of circle, so what is the center of circumvolve?
Given, (x-4)2+(y+7)2=9 is the equation of circle. If we compare this equation with the standard equation we get:
(x-h)2+(y-yard)2 = atwo
h=4 and y = -7
Therefore, (iv,-7) is the center of circumvolve
How do we know if an equation is the equation of circle?
If x and y are squared and the coefficient of ten2 and ytwo are same, so it is equation of circle. For case, 3x2+3y2 = 12 is a circle's equation.
Equation Of Circle General Form,
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