### Circles

Definition: A circle is the locus of a point which moves in a plane such that the distancem from a fixed point in that plane is always constant.
* The fixed point is called centre of the circle.
* The fixed distance is called radius of the circle.

EQUATION OF CIRCLE WITH CENTRE (h, k) AND RADIUS 'r'.
The equation of the circle with centre (h, k) and radius 'r' is (x - h)2 + (y - k)2 = r2.
* The general equation of the circle is x+ y2 + 2gx + 2fy + c = 0; where g, f, c constants and (-g, -f) is centre of the circle and  (g2 + f2 - c 0) is radius of the circle.
* A second degree equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a circle if
(i) a = b; coefficient of x = coefficient of y
(ii) h = 0; coefficient of xy = 0
(iii) ∆ = abc + 2fgh − af2 − bg2 − ch2 ≠ 0
(for point circle ∆ = 0)
(iv) g2 + f2 - c 0
* If two circles have same centre and different radius, then they are called concentric circles.
* The equations of concentric circles differ by only the constant term.

Equation of Circle on a given Diameter
Equation of circle having (x1, y1) and (x2, y2) as the extremities of the diameter is
(x - x1)(x - x2) + (y - y1)(y - y2) = 0

Different forms of the Equations of a Circle

1) Equation of the circle having centre (h, k) and which touches X - axis is (x - h)2 + (y - k)2 = k2

2) Equation of the circle having centre (h, k) and which touches Y-axis is (x - h)2 + (y - k)2 = h2

3) Equation of the circle having centre (h, h) and which touches both the axis is (x - h)2 + (y - h)2 = h2

4) Equation of the circle passing through origin (0, 0) and which cuts X - axis at (h, 0) and Y - axis at (0, k) is x2 + y2 - hx - ky = 0

Intercepts made by a Circle on the axes
Let the circle x2 + y2 + 2gx + 2fy + c = 0, then
X - intercept = AB =
and Y - intercept = CD =

Note: Let the circle equation S = x2 + y2 + 2gx + 2fy + c = 0 and P(x1, y1) be a point, then we define
S11 = x12 + y12 + 2gx1 + 2fy1 + c
S1 = xx1 + yy1 + g(x + x1) + f(y + y1) + c

POSITION OF A POINT WITH RESPECT TO A CIRCLE
Let the circle S = x2 + y2 + 2gx + 2fy + c = 0 and P(x1, y1) be any point, then
(i) if S11 > 0 then 'P' lies out side the circle.
(ii) if S11 < 0 then 'P' lies inside the circle.
(iii) if S11 = 0 then 'P' lies on the circle.

PARAMETRIC FORM OF CIRCLE
The parametric coordinates of any point on the circle (x - h)2 + (y - k)2 = r2 is (h + r cos , k + r sin ) where θ is a parameter and

* Let 'P' be any point. If we draw a secant line through the point 'P' and it cuts the circle at two different points A and B, then PA.PB is always constant and PA.PB = S11

INTERSECTION OF A LINE AND A CIRCLE
Let the equation of circle be x2 + y2 + 2gx + 2fy + c = 0
Equation of line be y = mx + c
Let the length of perpendicular from centre to the line = d and radius  = r, then
(i) if d > r  the line does not intersect the circle
(ii) if d < r  the line intersect the circle in two different points
(iii) if d = r  the line touches the circle at single point
i.e., the line is tangent to the circle

THE LENGTH OF THE INTERCEPT CUT OFF FROM A LINE BY A CIRCLE
The length of the intercept cut off from the line y = mx + c by the circle
The equation of tangent to the circle (x - h)2 + (y - k)2 = r2 at the point (h + r cos , k + r sin ) is (x - h)cos + (y - k)sin  = r

Pairs of Tangents
If a point 'P' lies out side the circle, then we can draw two tangents through the point 'P' to the circle.  Equation of pairs of tangents drawn from a point P (x1, y1) to the circle

LENGTH OF THE TANGENTS
The length of the tangent from the point P(x1, y1) to the circle
S ≡ x2 + y2 + 2gx + 2fy + c = 0 is

DIRECTOR CIRCLE: The locus of point of intersection of perpendicular tangents to
a given circle is known as director circle.
The equation of the director circle of the circle (x - h)2 + (y - k)2 = a2 is
(x - h)2 + (y - k)2 = 2a2

CHORD OF CONTACT: The chord joining the points of contact of the two tangents to a circle drawn from a given out side point is called the chord of contact of tangents.

TANGENT TO A CIRCLE
The tangent at a point 'P' is the limiting position of a secant PQ when 'Q' tends to 'P' along the circle and point 'P' is called the point of contact of the tangent.

DIFFERENT FORMS OF THE EQUATIONS OF TANGENTS
1) Slope form:
The equation of tangent of slope 'm' to the circle x2 + y2 = a2 is
and the point of contact is
* The equation of tangent of slope 'm' to the circle x2 + y2 + 2gx + 2fy + c = 0 are

2) Point form: The equation of tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at the point (x1, y1) is S1 = xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
3) Parametric form: The equation of the tangent to the circle x2 + y2 = a2 at the point(a cos , a sin ) is x cos  + y sin = a.
The equation of the chord contact of tangent drawn from a point (x1, y1) to the circle
S ≡ x2 + y2 + 2gx + 2fy + c = 0 is S1 = 0
xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0

CHORD BISECTED AT A GIVEN POINT
The equation of the chord of the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 bisected at a point (x1, y1) is given by S1 = S11.

NORMAL TO A CIRCLE AT A GIVEN POINT
The normal of a circle is a straight line passing through the given point and perpendicular to the tangent at the point of contact.
The normal always passing through the centre of the circle.

COMMON TANGENTS TO TWO CIRCLES
Direct common tangents: If the two circles on same side of the given tangent, then it is called direct common tangent.
Transverse common tangents: If the two circles on opposite side of given tangent then it is called transverse common tangent.

COMMON TANGENTS
Let the circles (x - h1)2 + (y - k1)2 = r12  ................... (1)
(x - h2)2 + (y - k2)2 = r2    .................... (2)
with centres C1(h1, k1) and C2 (h2, k2) and radii 'r1' and 'r2' respectively.
CASE - I : If  C1C2 > r1 + r2
In this case two circles do not intersect. Then two transverse common tangents and two direct common tangents exist and point of intersection of direct common tangents (B) and transverse common tangents (A) lies on line joining 'C1' and 'C2'
The point 'A' divides C1 and C2 in the ratio r1 : r2 internally. The point 'B' divides the line segment joining 'C1' and 'C2' in the ratio r1 : r2 externally.

CASE II: If C1C2 = r1 + r2
In this case two circles touching externally, then two direct common tangents and one transverse common tangent exists.

CASE III: If C1C2 < r1 + r2
In this case two circles are intersect at two different points, then only two direct common tangents exist.

CASE IV: If C1C2 = r1 − r2
In this case two circles touch internally (One lies inside other and touches it), then only one direct common tangent exists.

CASE V: If C1C1 < r1 − r2
In this case two circles are such that one lies inside other and do not touch one another, then no common tangent exists.

COMMON CHORD
The chord joining the points of intersection of two given circles is called their common chord.
* The equation of the common chord of two circles
S1 ≡ x2 + y2 + 2g1x + 2f1y + c1 = 0 and
S1 ≡ x2 + y2 + 2g2x + 2f2y + c2 = 0 is
S1 - S2 = 0
i.e., 2(g1 − g2) x + 2(f1 − f2)y + c1 − c2 = 0

Length of Common Chord
Length of Common Chord of two circles S1 ≡ 0 and S2 ≡ 0 is  where r1 is radius of the circle S≡ 0 and d1 is perpendicular distance from the centre of the circle S≡ 0 to the common chord.
* If the length of the common chord is zero, then the two circles is touch each other and the common chord becomes common tangent at their point of contact.
* The length of common chord of two circles will be of  maximum length if it is a diameter of the smaller of the two circles.

ANGLE OF INTERSECTION OF TWO CIRCLES
The angle of intersection of two circles is defined as the angle between the tangents to the two circles at their point of intersection.
If 'θ' is angle of intersection of two circles
S1 ≡ x2 + y2 + 2g1x + 2f1y + c1 = 0 and
S2 ≡ x2 + y2 + 2g2x + 2f2y + c2 = 0, then

Where r1 = radius of the circle S1 = 0
r2 = radius of the circle S2 = 0
and d = distance between the centres of the two circles.

ORTHOGONAL CIRCLES
Two circles are said to intersect orthogonally if their angle of intersection is a right angle.
The condition for orthogonality of two circles
S1 ≡ x2 + y2 + 2g1x + 2f1y + c1 = 0
and S2 ≡ x2 + y2 + 2g2x + 2f2y + c2 = 0 is 2[g1g2 + f1f2] = c1 + c2

The radical axis of two circles is the locus of a point which moves in such a way that the lengths of the tangents drawn from it to the two circles are equal.
The equation of Radical axis of two circles.
S1 ≡ x2 + y2 + 2g1x + 2f1y + c1 = 0
and S2 ≡ x2 + y2 + 2g2x + 2f2y + c2= 0
is S1 - S2 = 0 i.e., 2(g1 − g2)x + 2(f1− f2)y + c1− c2 = 0

When two circles are intersect ( ... r2 − r1 < C1C2 < r1 + r2), then the common chord coincide with the Radical axis ( ... The points lying on common chord lies on the Radical axis)
When one circle lies inside the other ( ... C1C2 < r1 − r2)
In this case Radical axis and common chord both do not exist.
* The Radical axis of two circles is always perpendicular to the line joining the centres of the circles.
* The Radical axis of three circles whose centres are non-collinear taken in pairs are concurrent.
* The locus of the centre of the circle cutting two given circles orthogonally is their Radical axis.

The point of intersection of Radical axis of three circles whose centres are non collinear taken in pairs is called their Radical centre.
The circles with centre at the Radical centre and radius equal to the length of the tangent from it to one of the circles intersects all the three circles orthogonally.

FAMILY OF CIRCLES
The equation of family of circles passing through the intersection of a circle
S ≡ x2 + y2 + 2gx + 2fy + c = 0 and line L = lx + my + n = 0 is
x2 + y2 + 2gx + 2fy + c + λ (lx + my + n) = 0 i.e., S + λL = 0 where λ  R.
The equation of the family of circles passing through the points A(x1, y1), B(x2, y2) is

The equation of the family of circles touching the circle
S ≡ x2 + y2 + 2gx + 2fy + c = 0 at a point (x1, y1) is
i.e., x2 + y2 + 2gx + 2fy + c + λ[xx1 + yy1 + g(x + x1) + f(y + y1) + c] = 0
The equation of the family of circles passing through the intersection of the circles
S1 ≡ 0 and S2 ≡ 0 is
(Where λ(≠ −1)  R)

CO - AXIAL SYSTEM OF CIRCLES
*  A system of circles is said to be a coaxial system of circles if each pair of the circles in the system has the same radical axis.
* Since the line joining the centres of two circles is perpendicular to their radical axis.
* Then the centres of all circles of a coaxial system lie on a straight line which is perpendicular to the common radical axis.
*  Circles passing through two fixed points 'P' and 'Q' form a co-axial system because every pair of circles has the same common chord PQ and therefore the same radical axis which is perpendicular bisector of 'PQ'.
If S ≡ 0 is the equation of a member of a system of coaxial circles and the equation of common radical axis L ≡ 0, then the equation S + λL = 0 (λ  R) represents the coaxial system of circles.

LIMITING POINTS
Limiting points of a coaxial system of circles are the members of the system which have zero radius.

Posted Date : 17-02-2021

గమనిక : ప్రతిభ.ఈనాడు.నెట్లో వచ్చే ప్రకటనలు అనేక దేశాల నుండి, వ్యాపారస్తులు లేదా వ్యక్తుల నుండి వివిధ పద్ధతులలో సేకరించబడతాయి. ఆయా ప్రకటనకర్తల ఉత్పత్తులు లేదా సేవల గురించి ఈనాడు యాజమాన్యానికీ, ఉద్యోగస్తులకూ ఎటువంటి అవగాహనా ఉండదు. కొన్ని ప్రకటనలు పాఠకుల అభిరుచిననుసరించి కృత్రిమ మేధస్సు సాంకేతికతతో పంపబడతాయి. ఏ ప్రకటనని అయినా పాఠకులు తగినంత జాగ్రత్త వహించి, ఉత్పత్తులు లేదా సేవల గురించి తగిన విచారణ చేసి, తగిన జాగ్రత్తలు తీసుకొని కొనుగోలు చేయాలి. ఉత్పత్తులు / సేవలపై ఈనాడు యాజమాన్యానికి ఎటువంటి నియంత్రణ ఉండదు. కనుక ఉత్పత్తులు లేదా సేవల నాణ్యత లేదా లోపాల విషయంలో ఈనాడు యాజమాన్యం ఎటువంటి బాధ్యత వహించదు. ఈ విషయంలో ఎటువంటి ఉత్తర ప్రత్యుత్తరాలకీ తావు లేదు. ఫిర్యాదులు తీసుకోబడవు.