Circle
A circle is locus of a variable point which moves in a plane such that its distance from a fixed point is constant.
Locos of a point $P( h, k )$ such that its distance from a fixed point $O_1( x_0, y_0 )$ is constant and is equal to a units where $a \ge 0$.
$\Rightarrow PO_1 = a$
$\Rightarrow \sqrt{( h – x_0)^2 + ( k – y_0 )^2} = a$
$ \Rightarrow ( x – x_0 )^2 + ( y – y_0 )^2 = a^2$ represent equation of a circle in standard form with center as $O_1 ( x_0, y_0 )$ and radius as a units.
Some of basic geometrical properties of circle.
| 1 | Equal chords subtend equal angle at the center of circle. |
| 2 | Equal chords are equidistant from the center of a circle. |
| 3 | The tangent drawn from external point to the circle are of equal length. |
| 4 | Perpendicular bisector of a chord always passes through the center of a circle and vise versa. |
| 5 | Greater the distance of the chord from center, smaller will be length of chord. Maximum length of chord is the diameter of a circle. |
| 6 | Angle subtended at the center is double the angle subtended on the arc on opposit segment. |
| 7 | Angle in the same segment are equal. |
| 8 | If two chords AB , CD of a circle intersects either inside or outside the circle, then area of rectangle contained by the part of one of the chord is equal to area of rectangle contained by the part of the other chord. |
Let AB and CD intersect each other at P then $ AP \times PB = PC \times PD$
Equation of circle in different forms.
(1). The equation of a circle with center as origin $ O ( 0, 0)$ and the radius as “$a$” unit is $x^2 + y^2 = a^2$.
(2). The equation of a circle with center as a point $O_1 ( \alpha , \beta) $ and radius as “$a$” units is $(x – \alpha)^2 + (y – \beta)^2 = 0$.
(3). The General equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0 $, where center and radius of the circle is $( g , f )$ and r=$\sqrt{g^2 + f^2 – c}$ respectively.
Where for a circle to be a real circle radius $r \ge 0$ hence $r = \sqrt{g^2+f^2-c} \ge 0$
(4). Equation of a circle with points $ A(x_1,y_1)$ and $ B(x_2,y_2) $ as extremities of a one of its diameter is $(x – x_1)(x – x_2) + (y – y_1)(y – y_2) = 0$
Note :-
The general second degree equation $ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a equation of a circle if
(1) Coefficent of $x^2$ = Coefficent of $y^2$ i.e $a = b$
(2) Coefficient of $xy$ = zero. i.e $h=0$.
Illustration :- Find center and radius of circle $2x^2 + 2y^2 – 4x – y -2 = 0 $.
Solution :- To find center and radius, first ensure that Coefficent of $x^2$ and $y^2$ be 1.
Hence divide the equation of circle throughout by 2 we get,
$x^2 + y^2 – 2x – \cfrac{1}{2}y – 1 = 0$
Now center is $O_1 ( -g,-f)$ = $(- \cfrac{1}{2}$Coefficent of x , – $\cfrac{1}{2}$Coefficent of y )
= $( 1, \cfrac{1}{4})$
And radius $r ={\sqrt{g^2+f^2-c}}\,=\,{\sqrt{1+\frac{1}{16}+1}}$
Hence Center is $O_1 = (1,\cfrac{1}{4})$ and radius $r = \cfrac{\sqrt{33}}{2}$
Equation of a circle in different condition.
Condition
Equation of Circel
Figure
(1) The circle that touches both coordinate axis, with center as $(a,a)$ and raidus as “$a$”.
$ ( x – a )^2 + (y – a)^2 = a^2 $
(2) The circle that touches the x – axis, with center as $(\alpha,a)$ and raidus as “$a$”.
$ ( x – \alpha )^2 + (y – a)^2 = a^2 $
$ ( x – \alpha )^2 + (y + a)^2 = a^2 $
(3) The circle that touches the y – axis, with center as $(a,\beta)$ and raidus as “$a$”.
$ ( x – a )^2 + (y -\beta)^2 = a^2 $
$ ( x + a )^2 + (y – \beta)^2 = a^2 $
(4) The circle with OB as one of its diameter where $O$ being the origin and coordinates of point $B(\alpha,\beta)$ is
$ x^2 + y^2 – \alpha x – \beta y = 0 $
Parametric equation of a circle.
For circle with center as origin and radius as 'a'
Let ‘a’ be the radius of a circle with center as origin $O\,(0,0)$. Let P be any point $P (x,y)$ on the circle, and radius OP makes an angle of $\theta$ with positive direction of x-axis. Then $\theta$ is the parameter.
Drop perpendicular PA on x- axis, with point A on x-axis.
$therefore $ in triangle $\Delta OPA$
$ OP = a $ units, and $OA = x$ also $AP = y$
$\therefore Cos\theta = \cfrac{OA}{OP}=\cfrac{x}{a}\,—-(1)$
and $Sin\theta = \cfrac{AP}{OP}=\cfrac{y}{a}\,—-(2)$
Hence from (1) and (2) we get,
$x\,=\,aCos\theta$ and $y=\,=\,aSin\theta$
Hence parametric equation of circle with center as origin and radius as ‘a’ units is $x\,=\,aCos\theta$ and $y=\,=\,aSin\theta$ where $\theta$ is a parameter and $\theta \in [0,2\pi)$
For circle with center as $O_1(\alpha,\beta)$ and radius 'a' units.
For a circle with center as $O_1(\alpha,\beta)$ and radius ‘a” units.
Equation of circle is $(x – \alpha)^2 + (y – \beta)^2 = a^2$
Let $ X = x – \alpha $ and $ Y = y – \beta$
$\therefore$ equation of circle is $X^2 + Y^2 = a^2$
$\therefore $ parametric equation is $X = x – \alpha=aCos\theta$ and $Y = y – \beta = aSin\theta$ where $\theta$ is a parameter, and $\theta \in [0,2\pi)$