Circles

Formulas are organized based on their importance in the CAT exam.
1 Length of Tangent from External Point
2 Power of a Point Through a Chord
3 Condition for Two Circles to Touch
4 Parametric Equations
5 Family of Circles Through Two Points
6 Circle Touching Coordinate Axes
7 Orthogonality of Two Circles
8 Circle as Locus of a Right Angle
9 Radical Axis of Two Circles
10 Tangent Length from an External Point
11 Common Tangents to Two Circles
12 Condition for a Line to Be Tangent to a Circle
13 Equation of a Circle Tangent to Two Given Lines
14 Circle Touching Both Coordinate Axes
15 Diameter Endpoints Form
16 Chord of Contact
17 Chord with Given Midpoint
18 Orthogonality Condition
19 Chord Length Using Central Angle
20 Radical Axis of Two Circles
21 Circle Touching the $$x$$-Axis at a Given Point
22 Euler's Formula for Incenter and Circumcenter
23 Circle with Centre at the Origin
24 General Equation of a Circle
25 Euler's Inequality
26 Distance Between Centre and a Line
27 Circle Touching a Line at a Given Point
28 Diameter of a Circle
29 Circle Through Origin in Centre-Radius Form
30 Circle in Polar Coordinates
31 Circle Tangent to the $$y$$-Axis and Passing Through a Point
32 Circle Tangent to the $$x$$-Axis
33 Tangent to Circle at Origin
34 Power of a Point
35 Common Chord of Two Circles with Equal Radii
36 Common Chord of Two Intersecting Circles
37 Condition for a Real Circle
38 Circle Through the Origin
39 Locus of Centres of Circles Passing Through Two Fixed Points
40 Circle Through Three Given Points
41 Locus of Centres of Circles Touching a Given Line
42 Radius from Diameter Endpoints
43 Locus of Points with Equal Distances from Two Fixed Points
44 Circle Tangent to the $$x$$-Axis and Passing Through a Point
45 Parametric Coordinates of a Circle
46 Conversion Between Cartesian and Polar Coordinates
47 Maximum Distance of a Point on a Circle from Another Point
48 Circle Passing Through the Origin with Centre $$(h,k)$$
49 Equation of a Circle with Diameter on a Given Line
50 Circle Touching a Given Circle Externally
51 Area of a Sector
52 Area of a Circular Segment
53 Centre of a Circle Tangent to Two Lines and Passing Through a Point
54 Condition for a Line to Be a Secant
55 Length of Chord Formed by a Line
56 Chord with Given Midpoint for a Circle at Origin
57 Polar of a Point
58 Circle Touching a Line at a Given Point
59 Tangent Length from a Point to a General Circle
60 Common Chord of Two Intersecting Circles
61 Radius Perpendicular to Tangent
62 Intersecting Chords Theorem
63 Pole of a Line
64 Pole of a Line for a General Circle
65 Normal to a Circle at a Point
66 Distance of a Chord from the Centre
67 Perpendicular from Centre Bisects a Chord
68 Concentric Circles
69 Equal Power Condition
70 Equal Chords and Equal Distances
71 Radical Axis of Two Concentric Circles
72 Standard Equation of a Circle
73 Condition for Two Circles to Intersect
74 General Equation of Circle
75 Standard Equation of Circle
76 Intersection of Tangents at Two Points
77 Length of Normal from a Point on a Circle
78 External Common Tangents
79 Centre of a Circle Tangent to Two Intersecting Lines
80 Circle Through the Intersection of Two Circles
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