Short Notes on Circle

Date: March 27, 2020 By: Ashok Kumar

1. DEFINITION

A circle is the locus of a point which moves in a plane such that its distance from a

fixed point is always constant. The fixed point is called the centre of the circle and the constant distance, the radius of the circle.

STANDARD EQUATION OF A CIRCLE

The equation of a circle with the centre at (a, b) and radius r, is given by (x – a)2 + (y – b)2 = r2 .If the centre of the circle is at the origin and the radius is r, then equation of circle is x2 + y2 = r2.

,a b

r

x

y

r

x

y

EQUATION OF CIRCLE IN DIFFERENT CONDITIONS

Condition Equation

(i) Touches both the axes with centre (a, a) and radius a

(x–a)2 + (y–a)2 = a2

(a, a)a

ax

y

(ii) Touches x–axis only with centre

(, a) and radius a

(x–)2 + (y–a)2 = a2

ax

y

( ,a)

(iii) Touches y–axis only with centre (a, ) and radius a

(x–a)2 + (y–)2 = a2

a

x

y

(a, )

(iv) Passes through the origin with centre

2,

2 and radius

4

22 .

x2 +a y2 – x – y =

0

x

y

( / 2, / 2 )

CONDITION FOR THE GENERAL EQUATION OF SECOND DEGREE IN X AND Y

TO REPRESENT A CIRCLE The general equation of second degree ax2+2hxy+by2+2gx+2fy+c=0 represent a circle ,if

Coefficient of x2=coefficient of y2 i.e. a=b Coefficient of xy=0 i.e. h=0.

GENERAL EQUATION OF A CIRCLE The general equation of a circle is of the form x2 + y2 + 2gx + 2fy + c = 0 where g, f and c are constants. The given equation of circle in the standard form can be written

as 2

2 2 2 2( ) ( )x g y f g f c . Hence the coordinates of its centre are (-g, -f)

and radius is 2 2g f c .

Case I: if 2 2g f c >0, a real circle is possible.

Case II: if 2 2g f c =0 , the circle is called a point circle.

Case III: if 2 2g f c < 0 no real circle is possible.

Working rule to find the centre and radius of a circle whose equation is given: STEP I: Make The coefficients of x2 and y2 equal to 1 and right hand side equal

to zero.

STEP II: The coordinate of the centre will be (a, b) where a=1

2 (coefficient of x)

and b=1

2 (coefficient of y).

STEP III: Radius= 2 2 constant term a b

EQUATION OF A CIRCLE WHOSE END POINTS OF ANY DIAMETER IS GIVEN

Equation of the circle with points P(x1, y1) and Q(x2, y2) as extremities of a diameter is given by (x – x1)(x – x2) + (y – y1)(y – y2) = 0.

INTERCEPT MADE BY THE CIRCLE ON THE AXIS

Let the equation of circle be x2 + y2 + 2gx + 2fy + c = 0………(1) X- INTERCEPT: Intercept made by the circle on the x-axis is called the X-intercept. The circle will intersect the x-axis where y = 0x2 + 2gx + c = 0 …(2) Here three cases arises

Case I: If discriminant > 0 i.e. 2g - c >0 circle will intersect the axis at two distinct

and real points let A (x1, 0) and B (x2, 0). Length of the intercept

2 2

2 1 1 2 1 2AB = x - x = x + x - 4x x = 2 g - c .

A(x

1, 0)

B(x2, 0)

x

y

Case II: If discriminant = 0 i.e. g2=c. Then the circle will touch the x-axis. In this case

length of the intercept made by the circle on the x-axis will be zero.

Case III: If discriminant < 0 i.e. 2g - c <0, in this case circle will neither touch nor

intersect the x-axis Y- INTERCEPT: Intercept made by the circle on the y-axis is called the Y-intercept. The circle will intersect the y-axis where x = 0y2 + 2fy + c = 0 …(2)

Again three cases arises

Case I: When discriminant > 0 i.e. 2f - c >0 circle will intersect the y-axis at two distinct and real points say A(0, y1) and B(0, y2). Length of the y-intercept

2 2

2 1 1 2 1 2AB = y - y = y + y - 4y y = 2 f - c

y

1A(0,y )

2B(0,y )

x

Case II: If discriminant = 0 i.e. 2f - c =0 Then the circle will touch the axis. In this case length of the intercept made by the circle on the x-axis will be zero.

Case III: If discriminant < 0, i.e. 2f - c <0 in this case circle will neither touch nor

intersect the axes.

PARAMETRIC EQUATION OF A CIRCLE Let the equation of circle be (x – a)2 + (y – b)2 = r2. Hence from the diagram it is clear that co-ordinates of any point on

the circle can be taken as (a + r cos,

b + r sin) where 0 < 2 .

x = a + r cos and y = b + r sin is called the parameter equation of the circle.

POSITION OF A POINT WITH RESPECT TO A CIRCLE

Let the equation of the given circle be x2+y2+2gx+2fy+c=0 and the given point be

P(,). Now, If the distance of the point P(,) from the centre of the circle(O) is greater

than the radius of the circle i.e. 2 2 > f PO g c , then the point will lie outside the

circle.

2 + 2 + 2g + 2f + c > 0, point P will lie outside the circle.

C

r

(a, b)

sinr

b

P (x, y)

x

y

r cosa

Similarly

If 2 + 2 + 2g + 2f + c = 0, point P will lie outside the circle.

If 2 + 2 + 2g + 2f + c <0 0, point P will lie outside the circle.

INTERSECTION OF A LINE WITH A CIRCLE

CONCEPT 1: Let S=0 be a circle with centre C and radius r ,L=0 be a line and p be perpendicular distance from the centre to the line L=0. Case I: If p>r i.e. the distance from the centre to the line is greater than the radius

then the circle doesn’t intersect the line. Case II: If p=r i.e. the distance from the centre to the line is equal to the radius then

the circle will intersect the line in one and only one point i.e. circle touches the line. This is also called the condition of tangency

Case III: If p<r i.e. the distance from the centre to the line is less than the radius then the circle intersect the line in two distinct points.

p r

p

r

p r

p

r

p r

p

r

Case I Case II Case III

EQUATION OF CHORD WHOSE MID POINT IS GIVEN

The equation of the chord of the circle S 0, whose mid point (x1, y1) is given by T = S1.

EQUATION OF TANGENTS

Definition of tangent: The tangent at any point of a circle if defined to be a straight line which meet the circle at that point but being produced, doesn’t cut it. The point where tangent meets the curve is called the point of contact.

Equation of the tangent to x2 + y2 + 2gx + 2fy + c = 0 at (x1 , y1) is given by xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0.

The condition that the straight line y = mx + c is a tangent to the circle x2 + y2=

a2 is c2 = a2 (1 + m2) and the point of contact is (–a2m/c, a2/c) i.e. y = mx 2m1a is always a tangent to the circle x2 + y2 = a2 whatever be the value of

m.

EQUATION OF TANGENT FROM ANY POINT OUTSIDE THE CIRCLE

METHOD 1: Let the equation of given circle be (x – a)2 + (y – b)2 = r2, and we have to find equation of tangent from the point P(x1, y1) outside the circle. Equation of any line passing through P(x1, y1) is given by y – y1 = m (x – x1). For different values of m we get different lines. For some particular value of m this equation will also represent a tangent to circle. To find the value of m apply the condition of tangency. METHOD 2: The joint equation of a pair of tangents drawn from the point P(x1, y1) to the

circle x2 + y2 + 2gx + 2fy + c = 0 is T2=SS1, where S x2+y2+2gx +2fy + c = 0 ,S1

x12+y1

2+2gx1+2fy1+c and T xx1+yy1+g(x+x1)+f(y +y1)+c

EQUATION OF NORMAL

The normal at any point of a curve is the straight line which is perpendicular to the tangent at the point of tangency (point of contact). Normal of any circle always passes through the centre of the circle. The equation of the normal to the circle x2+y2+2gx+2fy+c=0 at any point (x1, y1) lying

on the circle is 1 1

1 1

x x y y

x g y f

LENGTH OF THE TANGENT

The length of the tangent drawn from a point (x1, y1) outside the circle S 0, to

the circle is 1S .

INTERSECTION OF TWO CIRCLES

Let the equation of two circles be S1 = x2 +y2 2g1x+2f1y+c1=0 and S2

= x2 +y2 2g2x + 2f2y + c2 = 0 and let their centres are represented by O1 and O2 and their radii be r1 and r2 respectively.

O1 O

2

r1

r2

Q1

Q2

P Q

Trasverse common tagent

Direct common tagents

Case I: when both the circles are non-intersecting and one lies outside the other

It the distance between the centres of two given circles is greater than the sum of their radii i.e. O1O2 > r1 +r2. Then both the circles will be non intersecting. In this case there exist four common tangents to these two circles. Two direct common tangents and two transverse common tangent. Working Rule to find direct common tangent: Step I: First find the point of intersection of direct common tangents say Q, which

divides O1 O2 externally in r1 : r2

Step II: Write the equation of any line passing through Q (, ), i.e. y- = m (x-

)…….(1) Step III: Find the two values of m, using the fact that the length of the perpendicular

on (1) from the centre of one circle is equal to its radius. Step IV: Substitutes these values of ‘m’ in (1), the equation of the two direct common

tangents can be obtained. Working Rule to find transverse common tangent: To fine the equations of transverse common tangent first find the point of intersection of transverse common tangents say P, which divides O1O2 internally in r1:r2. Then follow the step 2, 3 and 4. Case II: If the distance between the centres of the given circle is equal to sum of

theirs radii. In this case both the circle will be touching each other externally.

In this case two direct common tangents are real and distinct while the transverse tangents are coincident.

O1 O2 =|r1+r2|

O1

O2

P

r1

r2

The point of contact P can be find by using the fact that it divides O1 O2

internally in r1 : r2 . Case III: It the distance between the centres of the given circles is equal to difference

of their radii i.e. |O1 O2| = |r1-r2|, both the circles touches each other internally.

In this case point of contact divides O1 O2 externally in r1 : r2.

O1

O2

P

In this case only one common tangent exist. Case IV: It the distance between the centres of two given circle is less then the sum

of their radii but greater then the difference of their radii i.e. |r1 - r2| < O1 O2 < r1 + r2, in this case both the circle will intersect at two real

and distinct points.

In this case there exist two direct common tangents.

O1

O2

P

(-g1 -f

1)

(-g2 -f

2)

ANGLE OF INTERSECTION OF TWO CIRCLES Angle of intersection of two circles is defined as the angle between the tangents at this

point of intersection. Here angle O1PO2 = -

2 2 2

1 2 1 2

1 2

O P O P O Ocos

2O PO P

21 2 1 1 2

2 2 2 2

1 1 1 2 2 2

2g g 2f f C Ccos

2 g f C g f C

2 2 2

1 2 1 2

1 2

r r O Ocos

2r r

ORTHOGONAL INTERSECTION OF TWO CIRCLES The two circles are said to intersect orthogonally if the angle between the tangents at their point of intersection is 900. The condition for two circles S1=O and S2=O to cut each other orthogonally is

1 2 1 2 1 22g g 2f f c c .

Note: It two circle are intersecting orthogonally the tangent to one circle at the point

of the intersection passes through the centre of other circle. Case V: It the distance between the centres is less than the difference of their radii,

i.e. |O1 O2| < |r1-r2|, in this case one circle will lie completely inside the other circle. Hence there will be no common tangent.

O1

O2

r

2. LOCATION OF A CIRCLE IN RELATION TO A CIRCLE

Let S1 x2 + y2 + 2g1x + 2f1y+ c1 = 0 and S2 x2 + y2 + 2g2x + 2f2y+ c1 = 0 be two circles. Let D be the discriminant for the quadratic equation in x (or y) obtained by eliminating y (or x) from the two equations of the circle. Then (i) they are two intersecting circles if D > 0 (ii) they are nonintersecting (no common points) if D < 0 (iii) they touch each other if D = 0 (iv) If D < 0, i.e., the circles are nonintersecting then

(a) S1 = 0 is outside S2 = 0 if S2 (-g1, –f1) > 0 or S1 (–g2, –f2) > 0; equivalently, AB > r1 + r2 where A, B are centres and r1, r2 are radii respectively.

(b) S1 = 0 is inside S2 = 0 if S2 (–g1, –f1) < 0; equivalently, AB < |r2 – r1| (v) If D = 0, i.e., then the circles touches each other

(c) externally if AB = r1 + r2

(d) internally if AB = r1 – r2

CHORD OF CONTACT

From a point P(x1, y1) out side the circle two tangents PA and PB can be drawn to the circle. The chord AB joining the points of contact A and B of the tangents from P is called the chord of contact of P(x1, y1) with respect to the circle. Its equation is given by T = 0.

B

(x1, y1)P

A

FAMILY OF CIRCLES

(i) If S x2 + y2 + 2gx + 2fy + c = 0 and S x2 + y2 + 2gx + 2fy + c = 0 are two intersecting circles, then family of circles passing through the point of

intersection of S and S’ is given by S + S = 0, (where is a parameter –1).

(ii) If S x2 + y2 + 2gx + 2fy + c = 0 is a circle which is intersected by the straight line L=lx+my+n=0 at two real and distinct points, then the equation of the family of circles passing through the point of intersection of circle and the given line is

given by S + L = 0.(where is a parameter).

(iii) The equation of a family of circles passing through two given points (x1, y1) and (x2, y2) can be written in the form.

(x – x1)(x – x2) + (y – y1)(y – y2) + 01yx1yx1yx

22

11 λ (where is a

parameter).

(iv) The equation of the family of circles which touch the line y – y1 = m(x – x1) at

(x1, y1) for any value of m is (x – x1)2 + (y – y1)2 + [(y – y1) –m(x – x1)] = 0. If m

is infinite, the equation is (x – x1)2 + (y – y1)2 + (x – x1) = 0.

RADICAL AXIS The radical axis of two circles is the locus of a point from which the tangent segments to the two circles are of equal length.

Equation to the Radical Axis

Consider S x2 + y2 + 2gx + 2fy + c = 0

and S' x2 + y2 + 2g'x + 2f'y + c' =0, then S-S’=0 represents the equation of the Radical

Axis to the two circles i.e. 2x(g – g) + 2y(f – f) + c – c = 0.

Note:

1. If S = 0 and S' = 0 intersect in real and distinct points then S – S' = 0 is the

equation of the common chord of the two circles.

Commonchord

(-g2, - f2)(-g1, - f1)

2. If S' = 0 and S = 0 touch each other, then S – S' = 0 is the equation of the

common tangent to the two circles at the point of contact.

• • • (-g2 -f2) (-g1 -f1)

Common tangent

3. To find the equation of the radical axis of two circles, first make the coefficients

of x2 and y2 in the equation of the two circles equal to unity. 4. The radical axis of two circles is perpendicular to the line joining their centres. 5. The radical axis of three circles taken in pairs meet at a point , called the radical

centre of the circles. Coordinates of radical centre can be found by solving the equations S1=S2=S3=0.

6. The radical centre of three circles described on the sides of a triangle as

diameters is the orthocentre of the triangle. 7. If two circles cut a third circle orthogonally, then the radical axis of the two

circles pass through the centre of the third circle. or

the locus of the centre of a circle cutting two given circles orthogonally is the radical axis of the two circles.

8. The radical axis of the two circles will bisect their common tangents.