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CirclesI

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www.sakshieducation.com
CIRCLES PART - I
Equation of A Circle:
The equation of the circle with centre C (h, k) and radius r is (x – h)
2
+ (y – k)
2
= r
2
.
Proof:
Let P(x
1
, y
1
) be a point on the circle. P lies in the circle
⇔
PC = r
⇔
2211
(xh)(yk)r
− + − =
⇔
(x
1
– h)
2
+ (y
1
– k)
2
= r
2
. The locus of P is (x – h)
2
+ (y – k)
2
= r
2
.
∴
The equation of the circle is (x–h)
2
+ (y–k)
2
= r
2
.------(1)
Note:
The equation of a circle with centre srcin and radius r is (x–0)
2
+ (y–0)
2
= r
2
i.e., x
2
+ y
2
= r
2
which is the standard equation of the circle.
Note:
On expanding equation (1), the equation of a circle is of the form x
2
+ y
2
+ 2gx + 2fy + c = 0.
Theorem: If g
2
+ f
2
– c
≥≥≥≥
0, then the equation x
2
+ y
2
+ 2gx + 2fy + c = 0 represents a circle with centre (–g, –f) and radius
22
gfc
+ −
. Note:
If ax
2
+ ay
2
+ 2gx + 2fy + c = 0 represents a circle, then its centre =
gf ,aa
− −
and its radius
22
gfac|a|
+ −
.
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Theorem:
The equation of a circle having the line segment joining A(x
1
, y
1
) and B(x
2
, y
2
) as diameter is
1212
(xx)(xx)(yy)(yy)0
− − + − − =
. Let P(x,y) be any point on the circle. Given points A(x
1
, y
1
) and B(x
2
, y
2
). Now
2
APB
π
=
. (Angle in a semi circle.) Slope of AP. Slope of BP =-1
( )( ) ( )( )( )( ) ( )( )
121221212121
10
y y y y x x x x y y y y x x x x x x x x y y y y
− −
⇒
= −− −
⇒
− − = − − −
⇒
− − + − − =
Definition:
Two circles are said to be concentric if they have same center. The equation of the circle concentric with the circle x
2
+ y
2
+ 2gx + 2fy + c = 0 is of the form x
2
+ y
2
+ 2gx + 2fy + k = 0. The equation of the concentric circles differs by constant only.
Parametric Equations of A Circle: Theorem:
If P(x, y) is a point on the circle with centre C(
α
,
β
) and radius r, then x =
α
+ r cos
θ
, y =
β
+ r sin
θ
where 0
≤θ
< 2
π
.
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Note:
The equations x =
α
+ r cos
θ
, y =
β
+ r sin
θ
, 0
≤θ
< 2
π
are called parametric equations of the circle with centre (
α
,
β
) and radius r.
Note:
A point on the circle x
2
+ y
2
= r
2
is taken in the form (r cos
θ
, r sin
θ
). The point (r cos
θ
, r sin
θ
) is simply denoted as point
θ
.
Theorem:
(1) If g
2
-c > 0 then the intercept made on the x axis by the circle x
2
+ y
2
+ 2gx + 2fy + c = 0 is
2
2
g ac
−
2) If f
2
– c >0 then the intercept made on the y axis by the circle x
2
+ y
2
+ 2gx + 2fy + c = 0 is
2
2
f bc
−
Note:
The condition for the x-axis to touch the circle x
2
+ y
2
+ 2gx + 2fy + c = 0 (c > 0) is g
2
= c.
Note:
The condition of the y-axis to touch the circle x
2
+ y
2
+ 2gx + 2fy + c = 0 (c > 0) is f
2
= c.
Note:
The condition for the circle x
2
+ y
2
+ 2gx + 2fy + c = 0 to touch the coordinate axes is g
2
= f
2
= c.
Position of Point:
Let S = 0 be a circle and P(x
1
, y
1
) be a point I in the plane of the circle. Then i) P lies inside the circle S = 0
⇔
S
11
< 0 ii) P lies in the circle S = 0
⇔
S
11
= 0 iii) P lies outside the circle S = 0
⇔
S
11
= 0
Power of a Point:
Let S = 0 be a circle with centre C and radius r. Let P be a point. Then CP
2
– r
2
is called power of P with respect to the circle S = 0.
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Theorem:
The power of a point P(x
1
, y
1
) with respect to the circle S = 0 is S
11
.
Theorem:
The length of the tangent drawn from an external point P(x
1
, y
1
) to the circle S = 0 is
11
S
.
Very Short Answer Questions
1. Find the equation of the circle with centre C and radius r where. i) C = (1, 7), r = Sol.
Equation of the circle is ( x-h)
2
+ (y-k)
2
=
2
⇒
( x-1)
2
+ (y-7)
2
=
2
⇒
x
2
– 2x + 1 + y
2
– 14y + 49 =
⇒
x
2
+ y
2
– 2x – 14y + 0
⇒
4x
2
+ 4y
2
– 8x – 56y + 175 = 0
ii) C = (a, -b); r = a + b
Equation of the circle is (x-h)
2
+ (y-k)
2
=
2
Equation of the circle is ( x-a)
2
+ (y-(-b))
2
=
2
⇒
x
2
– 2xa + a
2
+ y
2
+2by + b
2
= a
2
+2ab + b
2
⇒
x
2
+ y
2
– 2xa + 2by – 2ab = 0
2. Find the equation of the circle passing through the srcin and having the centre at (-4, -3). Sol
. Centre (h, k) = (-4, -3) Equation of the circle is (x – h)
2
+ ( y –k)
2
= r
2
; (x +4)
2
+ (y +3)
2
= r
2
Circle is passing through srcin
∴
(0 +4)
2
+ (0 + 3)
2
= r
2

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