Circles

Formulas are organized based on their importance in the CAT exam.
1 Circle Through the Origin
2 Length of Tangent from External Point
3 Length of Normal from a Point on a Circle
4 External Common Tangents
5 Circle Through the Intersection of Two Circles
6 Condition for Two Circles to Touch
7 Parametric Equations
8 Power of a Point Through a Chord
9 Family of Circles Through Two Points
10 Circle Touching Coordinate Axes
11 Orthogonality of Two Circles
12 Circle as Locus of a Right Angle
13 Condition for a Line to Be Tangent to a Circle
14 Radical Axis of Two Circles
15 Equation of a Circle Tangent to Two Given Lines
16 Common Tangents to Two Circles
17 Tangent Length from an External Point
18 Circle Touching Both Coordinate Axes
19 Diameter Endpoints Form
20 Chord with Given Midpoint
21 Chord of Contact
22 Orthogonality Condition
23 Chord Length Using Central Angle
24 Radical Axis of Two Circles
25 Circle Touching the $$x$$-Axis at a Given Point
26 Euler's Formula for Incenter and Circumcenter
27 Circle with Centre at the Origin
28 General Equation of a Circle
29 Euler's Inequality
30 Distance Between Centre and a Line
31 Circle Touching a Line at a Given Point
32 Diameter of a Circle
33 Circle in Polar Coordinates
34 Circle Tangent to the $$y$$-Axis and Passing Through a Point
35 Condition for a Real Circle
36 Common Chord of Two Intersecting Circles
37 Circle Through Three Given Points
38 Locus of Centres of Circles Touching a Given Line
39 Radius from Diameter Endpoints
40 Locus of Points with Equal Distances from Two Fixed Points
41 Circle Tangent to the $$x$$-Axis and Passing Through a Point
42 Conversion Between Cartesian and Polar Coordinates
43 Circle Passing Through the Origin with Centre $$(h,k)$$
44 Equation of a Circle with Diameter on a Given Line
45 Locus of Centres of Circles Passing Through Two Fixed Points
46 Area of a Circular Segment
47 Circle Touching a Given Circle Externally
48 Condition for a Line to Be a Secant
49 Circle Through Origin in Centre-Radius Form
50 Length of Chord Formed by a Line
51 Tangent to Circle at Origin
52 Chord with Given Midpoint for a Circle at Origin
53 Polar of a Point
54 Power of a Point
55 Common Chord of Two Circles with Equal Radii
56 Circle Touching a Line at a Given Point
57 Radius Perpendicular to Tangent
58 Intersecting Chords Theorem
59 Tangent Length from a Point to a General Circle
60 Common Chord of Two Intersecting Circles
61 Pole of a Line
62 Pole of a Line for a General Circle
63 Normal to a Circle at a Point
64 Distance of a Chord from the Centre
65 Perpendicular from Centre Bisects a Chord
66 Concentric Circles
67 Equal Power Condition
68 Equal Chords and Equal Distances
69 Radical Axis of Two Concentric Circles
70 Standard Equation of a Circle
71 Condition for Two Circles to Intersect
72 General Equation of Circle
73 Standard Equation of Circle
74 Circle Tangent to the $$x$$-Axis
75 Maximum Distance of a Point on a Circle from Another Point
76 Parametric Coordinates of a Circle
77 Area of a Sector
78 Centre of a Circle Tangent to Two Lines and Passing Through a Point
79 Centre of a Circle Tangent to Two Intersecting Lines
80 Intersection of Tangents at Two Points
81 Circle Tangent to the $$y$$-Axis
82 Circle with Given Diameter as a Locus
83 Equation of Circumcircle Using Triangle Vertices
84 Circle Tangent to Both Axes and Passing Through a Point
85 Chord Subtending a Right Angle at the Centre
86 Area of a Circle
87 Chord Through the Centre
88 Family of Circles with Same Centre
89 Completing Square in Circle Equation
90 Circle with Centre on a Given Line
91 Condition for a Line to Be Tangent
92 Semiperimeter of a Triangle
93 Centre of a Circle Tangent to Another Circle
94 Area in Terms of an Exradius
95 Chord of a Circle Parallel to a Given Line
96 Circle Tangent to Two Parallel Lines
97 Circle Tangent to a Circle and Passing Through a Fixed Point
98 Angle Subtended by a Diameter
99 Condition for Separate Non-Intersecting Circles
100 Circle Touching the $$y$$-Axis at a Given Point
101 Common Tangent at the Point of Contact
102 Centre of the Circle Through Three Points
103 Distance of Radical Axis from the First Centre
104 Chord of Contact from an External Point
105 Circumference of a Circle
106 Equal Tangents from an External Point
107 Nine-Point Circle Radius
108 Centre from Diameter Endpoints
109 Centre of a Circle Internally Tangent to Another Circle
110 Number of Tangents from a Point
111 Equation of a Circle with Centre and Tangency Point
112 Circle Through a Fixed Point with Variable Centre
113 Locus of Centres of Circles Touching Two Parallel Lines
114 Circle Touching a Given Circle Internally
115 Imaginary Circle
116 Circle Through a Point and Tangent to a Coordinate Axis
117 Polar of a Point for Circle at Origin
118 Pair of Tangents from an External Point
119 Position of Point Relative to Circle
120 Maximum Chord of a Circle
121 Circle Through a Given Point
122 Equation of a Circle with Centre on the $$y$$-Axis
123 Angle of Intersection of Two Circles
124 Slope of Tangent and Normal
125 Exradius Opposite Side $$b$$
126 Condition for Line Tangent to Circle
127 Condition for Two Circles to Have a Common Tangent at an Intersection Point
128 Midpoint of a Chord
129 Secant-Secant Theorem
130 Equation of a Circle Touching a Given Line
131 Internal Common Tangents
132 Incircle Radius from Triangle Area
133 Condition for a Line to Not Meet a Circle
134 Tangent-Secant Theorem
135 Exradius Opposite Side $$c$$
136 Circle Passing Through Two Points and Tangent to a Line
137 Tangent-Chord Relation
138 Chord Subtending a Right Angle at the Circumference
139 Condition for a Point to Lie on a Circle
140 Exradius Opposite Side $$a$$
141 Area in Terms of Inradius
142 Circle Through Two Fixed Points with Centre on a Given Line
143 Heron's Formula
144 Distance Between Two Circles
145 Condition for External Tangency of Two Circles
146 Minimum Distance of a Point from a Circle
147 Angle Between Two Tangents
148 Tangent to a Circle at a Parametric Point
149 Angle Between Two Tangents and Central Angle
150 Tangent in Slope Form
151 Orthogonal Circles as a Special Case
152 Radius of a Circle Tangent to Two Lines
153 Radical Centre of Three Circles
154 Circle Tangent to Three Given Lines
155 Circle Orthogonal to a Given Circle
156 Condition for a Point to Lie Outside a Circle
157 Distance Between Circumcentre and Orthocentre
158 Centre of a Circle from General Equation
159 Area of Triangle Formed by Two Radii and a Chord
160 Polar of a Point on the Circle
161 Point Circle
162 Family of Circles Through the Common Points of Two Circles
163 Distance Between Nested Circles
164 Circle Passing Through Two Given Points
165 Equation of Chord with Given Midpoint
166 Circle as a Locus of Constant Distance
167 Equation of a Circle with Centre on the $$x$$-Axis
168 Centre of a Circle Tangent to Two Parallel Lines
169 Circle Through the Midpoints of the Sides
170 Radius of Circumcircle of a Right Triangle
171 Tangency Condition for a Line in Slope Form
172 Circle Through the Intersection of a Circle and a Line
173 Circle Tangent to Both Coordinate Axes
174 Length of Common Chord
175 Circle Equation After Translation of Origin
176 Chord of Contact for a Circle at Origin
177 Coaxal System of Circles
178 Power of a Point Using Circle Equation
179 Length of Common Chord of Two Intersecting Circles
180 Circumradius from Triangle Area
181 Radius of a Circle from General Equation
182 Condition for a Point to Lie Inside a Circle
183 Centre of Nine-Point Circle
184 Circle Through Origin and Tangent to a Given Line
185 Normal to Circle
186 Circle with a Given Centre Passing Through a Given Point
187 Pole-Polar Reciprocity
188 Number of Common Tangents
189 Condition for Internal Tangency of Two Circles
190 Chord of Contact as Polar
191 Condition for One Circle Inside Another Without Touching
192 Locus of Centres of Circles Tangent to a Fixed Circle
193 Tangent to a Circle at a Point in Standard Form
194 Intersection of a Circle and a Line
195 Radical Axis Perpendicular to Line of Centres
196 Length of an Arc
197 Length of Chord Cut by a Line
198 Circle Having a Given Line as Diameter
199 Tangent to a Circle at a Given Point
200 Inradius Using Heron's Formula
201 Tangent at Point on Circle