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0
INTERMEDIATE
MATHEMATICS – 2B
CIRCLE
IMPORTANT
DEFINITIONS
FORMULAE
IMP
QUESTIONS
NAME : ……………………………
GROUP : ……………………………
ROLL
...
GOD is great
1
Matrices
CIRCLE
(2 + 2 + 4 + 7 + 7 = 22)
1
...
The fixed point C is called the centre of the
circle
...
Various forms of a Circle
2
...
Centre C = (-g, -f )
Radius ( r ) = √𝑔2 + 𝑓 2 + 𝑐
3
...
Centre C = ( 0, 0)
Radius ( r ) = r
4
...
Centre C = ( h,k )
Radius ( r ) = r
𝐁𝐚𝐧𝐝𝐚𝐫𝐮 𝐂𝐡𝐢𝐫𝐚𝐧𝐣𝐞𝐞𝐯𝐢
𝐌
...
The general equation of second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 where the coefficients a, h, b, g, f
and c are real numbers , represents a circle if and only if
i
...
iii
...
Point circle :
If the radius of a circle is 0 then it is called a point circle
...
The equation of a point circle having the centre at the origin is x2 + y2 = 0
...
Unit circle :
If the radius of a circle is 1 then it is called a unit circle
...
Concentric circles :
Two or more circles are said to be concentric if their centres are same ( equal )
...
9
...
( since y – coordinate of the centre is zero )
10
...
𝐁𝐚𝐧𝐝𝐚𝐫𝐮 𝐂𝐡𝐢𝐫𝐚𝐧𝐣𝐞𝐞𝐯𝐢
𝐌
...
The intercept made on the X – axis by the circle x2 + y2 + 2gx + 2fy + c = 0 is 2√𝑔2 − 𝑐
...
The intercept made on the Y – axis by the circle x2 + y2 + 2gx + 2fy + c = 0 is 2√𝑓 2 − 𝑐
...
Secant of a circle :
If A and B are two distinct points on a circle then the line ⃡𝐴𝐵 through A and B is called a secant
...
Chord of a circle :
If A and B are two distinct points on a circle then the segment 𝐴𝐵 , the join of A and B is called a chord
...
15
...
16
...
17
...
Concyclic points :
P (𝑥1 , 𝑦1 ) , 𝑄(𝑥2 , 𝑦2 ) , R(𝑥3 , 𝑦3 ) and S(𝑥4 , 𝑦4 ) are said to be concyclic if these points lie on the same
circle
...
,
Maths - 2B
Circle
4
Matrices
Parametric equations of a circle :
19
...
20
...
21
...
22
...
𝑆1 = xx1 + yy1 + g( x + x1 ) + f( y + y1 ) + c
𝑆2 = xx2 + yy2 + g( x + x2 ) + f(y + y2 ) + c
𝑆11 = x12 + y12 + 2gx1 + 2fy1 + c
𝑆12 = x1x2 + y1y2 + g( x1 + x2 ) + f(y1 + y2 ) + c
...
Position of a point with respect to a circle :
A circle in a plane divides the plane into three parts namely
( i ) the interior of the circle ( lies in the circle )
( ii ) the circumference which is the circular curve ( lies on the circle )
(iii ) the exterior of the circle ( out side the circle )
24
...
Then
( i ) P lies in the interor of the circle ⟺ s11 < 0
( ii ) P lies on the circle ⟺ s11 = 0
(iii ) P lies in the exterior of the circle ⟺ s11 > 0
𝐁𝐚𝐧𝐝𝐚𝐫𝐮 𝐂𝐡𝐢𝐫𝐚𝐧𝐣𝐞𝐞𝐯𝐢
𝐌
...
Tangent of a circle :
A tangent to a circle is a straight line which touches
the circle at only one point
...
The tangent to a circle is
perpendicular to the radius at the point of
tangency
...
Length of tangent :
If P is an external point to the circle S = 0 and PT
is the tangent from P to the circle S = 0 then 𝑃𝑇 is
called the length of the tangent from P to the circle
...
So, PA and PB are the
lengths of tangent to the circle from an external
point P
...
If S = 0 is a circle and P (𝑥1 , 𝑦1 ) is an external point with respect to S = 0 then the length of the tangent
from P (𝑥1 , 𝑦1 ) to S = 0 is √𝑆11
...
Power of a point :
Suppose S = 0 is the equation of a circle with centre C and radius r
...
Then CP2 – r2 is called the power of point p with respect to S = 0
...
The power of a point P (𝑥1 , 𝑦1 ) with respect to the circle S = 0 is S11
...
The equation of a tangent to the circle x2 + y2 = r2 having slope m is y = mx ± √1 + 𝑚2
...
The equation of a tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 having slope m is
(𝑦 + 𝑓) = m(𝑥 + 𝑔) ± r √1 + 𝑚2
...
The equation of a tangent at 𝜃 of the circle x2 + y2 + 2gx + 2fy + c is (𝑥 + 𝑔) cos 𝜃 + (𝑦 + 𝑓) sin 𝜃 = r
...
The equation of a tangent at 𝜃 of the circle x2 + y2 = r2 is xcos𝜃 + ysin𝜃 = r
...
,
Maths - 2B
Circle
6
Matrices
34
...
The equation of a tangent at the point P (𝑥1 , 𝑦1 ) to the circle x2 + y2 = r2 is x2 + y2 - r2 = 0
...
normal of a circle :
The normal of a circle at any point is a straight line which is perpendicular to the tangent at the point of
contact
...
37
...
38
...
39
...
40
...
,
𝜃1 + 𝜃2
2
𝜃1 − 𝜃2
) = rsin (
2
)
...
The tangential condition for the line ax + by + c = 0 to the circle S = 0 is r = d
...
If d is the perpendicular distance from the centre of the cicle and r is the radius , the length of the chord
is 2 √𝑟 2 − 𝑑2
...
the tangential condition for the line
( i ) y = mx + c and the circle x2 + y2 = r2 is c2 = r2 ( 1+ m2 )
...
44
...
( ii ) the Y – axis is c = f2
( iii ) both the axes is c = g2 = f2
...
If 𝜃 is the angle between the tangents through a point P (𝑥1 , 𝑦1 ) to the circle S = 0 then tan 2 =
𝑟
√𝑠11
...
chord of contact :
If the tangents drawn from an external point P (𝑥1 , 𝑦1 ) to a circle S = 0 touch the circle at the points A and
B then the secant ⃡𝐴𝐵 is called the chord of contact of P with respect to S = 0
...
If P (𝑥1 ,
𝑦1 ) is an exterior point to the circle x2 + y2 + 2gx + 2fy + c = 0 then the equation of the chord of
contact of P with respect to S = 0 is S1 = 0
...
Pole and Polar :
Let S = 0 be a circle and P be any point in plane other than the centre of S = 0
...
𝐁𝐚𝐧𝐝𝐚𝐫𝐮 𝐂𝐡𝐢𝐫𝐚𝐧𝐣𝐞𝐞𝐯𝐢
𝐌
...
The equation of the polar of 𝑃(𝑥1 ,
𝑦1 ) with respect to S = 0 is S1 = 0
...
The pole of lx + my + n = 0 with respect to the circle x2 + y2 = a2 is (
−𝑎2 𝑙
𝑛
,
−𝑎2 𝑚
𝑛
)
...
The pole of lx + my + n = 0 with respect to the circle x2 + y2 + 2gx + 2fy + c = 0 is
(−𝑔 +
𝑙𝑟 2
lg + 𝑚𝑓−𝑛
, −𝑓 +
𝑚𝑟 2
lg + 𝑚𝑓−𝑛
)
...
Conjugate points :
Two points P and Q are said to be conjugate points with respect to the circle S = 0 , if Q lies on the polar of P
...
The condition that the points P (𝑥1 , 𝑦1 ) 𝑎𝑛𝑑 𝑄(𝑥2 , 𝑦2 ) are conjugate points with respect to the circle S = 0
is S12 = 0
...
Conjugate lines :
If P and Q are conjugate points with respect to the circle S = 0 then the polars of P and Q are called conjugate
lines with respect to the circle S = 0
...
The condition for the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 are conjugate with respect to the circle
x2 + y2 + 2gx + 2fy + c = 0 is r2 ( l1l2 + m1m2 ) = (l1g + m1f – n1 ) ( l2g + m2f – n2 )
...
The condition for the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 are conjugate with respect to the circle
x2 + y2 = r2 is r2 ( l1l2 + m1m2 ) = n1n2
...
Inverse points with respect to a circle :
Let S = 0 be a circle with centre C and radius r
...
CQ = r2
...
𝐁𝐚𝐧𝐝𝐚𝐫𝐮 𝐂𝐡𝐢𝐫𝐚𝐧𝐣𝐞𝐞𝐯𝐢
𝐌
...
If
P (𝑥1 , 𝑦1 ) is the mid point of a chord AB ( other than the diameter ) of the circle S = 0 then the equation of
secant ⃡𝐴𝐵 is 𝑠1
= 𝑠11
...
Common tangent of a circles :
A straight line L is said to be a common tangents to the circles S= 0 and S' = 0 if it is tangent
to both S = 0 and S' = 0
...
Touch each other of two circles :
Two circles are said to be touching each other if they have only one common tangent
...
,
Maths - 2B
Circle
10
Matrices
61
...
62
...
,
Maths - 2B
Circle
11
Matrices
63
...
the point of intersection of direct common tangents of
S = 0 and S' = 0 is called external centre of similitude
...
Internal centre of similitude :
Let S = 0 and S' = 0 be two circles
...
65
...
Furthur let 𝐶1 𝐶2 represents the line segment C1 to C2
...
( i ) Case – 1 : 𝒄𝟏 𝒄𝟐 > 𝒓𝟏 + 𝒓𝟐
In this case the two circles do not intersect and one circle will be away from the
other
...
of direct common tangents = 2
No
...
of common tangents = 4
𝐁𝐚𝐧𝐝𝐚𝐫𝐮 𝐂𝐡𝐢𝐫𝐚𝐧𝐣𝐞𝐞𝐯𝐢
𝐌
...
No
...
of transverse common tangents = 1
Total No
...
No
...
of transverse common tangents = 0
Total No
...
,
Maths - 2B
Circle
13
Matrices
( iv ) Case – 4 : 𝒄𝟏 𝒄𝟐 = |𝒓𝟏 − 𝒓𝟐 |
In this case the two circles touch each other internally
...
of direct common tangents = 1
No
...
of common tangents = 1
( v ) Case – 5 : 𝒄𝟏 𝒄𝟐 < |𝒓𝟏 − 𝒓𝟐 |
In this case the two circles do not intersect/touch and one circle will be
completely inside the other
...
of direct common tangents = 0
No
...
,
Maths - 2B
Circle
14
Matrices
Total No
...
66
...
𝐁𝐚𝐧𝐝𝐚𝐫𝐮 𝐂𝐡𝐢𝐫𝐚𝐧𝐣𝐞𝐞𝐯𝐢
𝐌
...
Find the equation of the circle passing through (1, 1) , (2, 1), (3, 2)
...
Find the equation of the circle passing through (3, 4) , (3, 2), (1, 4)
...
Find the equation of the circle passing through (5, 7) , (8, 1), (1, 3)
...
Find the equation of the circle passing through (2, 1) , (5, 5), (−6, 7)
...
Find the equation of the circle passing through (1, 2) , (3, −4), (1, 3)
...
If (2, 0), (0, 1) (4, 5) 𝑎𝑛𝑑 (0, 𝐶) are concyclic then find C
...
Show that the following four points in each of the following are concyclic and find the equation of the
circle on which they lie
...
(1, 1), (−6, 0), (−2, 2) , (−2, −8)
(ii)
...
(1, 2), (3, −4), (5, −6) , (19, 8)
(iv)
...
MODEL – 3
8
...
9
...
10
...
𝐁𝐚𝐧𝐝𝐚𝐫𝐮 𝐂𝐡𝐢𝐫𝐚𝐧𝐣𝐞𝐞𝐯𝐢
𝐌
...
Show that the circles x2 + y2 – 6x – 2y + 1 = 0 , x2 + y2 + 2x – 8y + 13 = 0 touch each other
...
12
...
Find the point of
contact and the equation of common tangent at their point of contact
...
Show that the circles x2 + y2 – 4x – 6y - 12 = 0 , x2 + y2 + 6x + 18y + 26 = 0 touch each other
...
14
...
Find the
point of contact
...
Find the direct common tangents of the circles x2 + y2 + 22x – 4y – 100 = 0 and x2 + y2 – 22x + 4y + 100 = 0
...
Find the transverse common tangents of the circles x2 + y2 – 4x – 10y + 28 = 0 and x2 + y2 + 4x – 6y + 4 = 0
...
The combined equation of the pair of tangents drawn from an external point P(𝑥1 , 𝑦1 ) to the circle S = 0
is SS11 = S12
...
Find the equations of circles which touch 2x – 3y + 1 = 0 at (1, 1 ) and having radius √13
...
Find the equation of circle which touch the circle x2 + y2 – 2x – 4y – 20 = 0 externally at (5, 5) with radius 5
...
Find the equation of circle which touch the circle x2 + y2 – 4x + 6y – 12 = 0 internally at (-1, 1) with radius 2
...
,
Maths - 2B
Circle
17
Matrices
IMP SHORT ANSWER TYPE QUESTIONS ( 4 MARKS QUESTIONS )
MODEL – 1
1
...
2
...
3
...
MODEL – 2
4
...
5
...
6
...
MODEL – 3
7
...
8
...
9
...
Find the condition that the tangents drawn from (0, 0) to S≡ 𝑥 2 + 𝑦 2 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 be
Perpendicular to each other
...
Find the length of the chord intercepted by the circle x2 + y2 – x + 3y – 22 = 0 on the line y = x – 3
...
Find the length of the chord intercepted by the circle x2 + y2 – 8x – 2y – 8 = 0 on the line x + y + 1 = 0
...
Find the length of the chord formed by x2 + y2 = a2 on the line xcos𝛼 + ysin𝛽 = p
...
,
Maths - 2B
Circle
18
Matrices
13
...
14
...
15
...
16
...
( Do the problem taking the line x + 2y – 8 = 0 instead of x + y – 8 = 0 )
17
...
18
...
19
...
20
...
Also find the equation of tangent
Parallel to it
...
AP March 2015
MODEL – 6
21
...
22
...
and find the other end of diameter
through A
...
Show that A(3, -1) lies on the circle x2 + y2 - 2x + 4y = 0
...
24
...
Show that the tangent at (-1, 2) of the circle x2 + y2 – 4x – 8y + 7 = 0 touches the circle
x2 + y2 + 4x + 6y = 0
...
BOARD MODEL PAPER
25
...
26
...
𝐁𝐚𝐧𝐝𝐚𝐫𝐮 𝐂𝐡𝐢𝐫𝐚𝐧𝐣𝐞𝐞𝐯𝐢
𝐌
...
Show tha the lines 2x + 3y + 11 = 0 and 2x – 2y – 1 = 0 are conjugate with respect to the circle
x2 + y2 + 4x + 6y – 12 = 0
...
TS May - 2015
Miscellaneous problems
28
...
29
...
30
...
31
...
32
...
𝐁𝐚𝐧𝐝𝐚𝐫𝐮 𝐂𝐡𝐢𝐫𝐚𝐧𝐣𝐞𝐞𝐯𝐢
𝐌