## Circles: Formulas, Examples and Worksheets with detailed theory and explanations

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*C**ramer (1750 A.D.) made formal use of the two axes and gane the eqution of a circle as*

*(y – a)*

^{2}+ ( b – x)^{2}= r.r. He gave the best exposition of the analytic geometry of his time.*K*

*ochanski gives an approximate method to find the length of the circumference of a circle.*Jones introduces the Greek latter to represent the ratio of the circumference of a circle to its diameter in his Synopsis palmariorum matheseos (A new introduction to Mathematics).

*F*

*euerbach publishes his discoveries on the nine point circle of a triangle.*

*N*

*icholas of Cusa studies geometry and logic. He contributes to the study of infinity, the infinitely large and the infinitely small. He looks at the circle as the limit of regular polygons.*

*Basic information about circles*

*A circle is all points in the same plane that lie at an equal distance from a center point. The circle is only composed of the points on the border. You could think of a circle as a hula hoop. It’s only the points on the border that are the circle. The points within the hula hoop are not part of the circle and are called interior points.**By Distance Formula we can find the distance between the midpoint and the circle border which is called the radius. A line segment that has the endpoints on the circle and passes through the midpoint is called the diameter. The diameter is twice the size of the radius. A line segment that has its endpoints on the circular border but does not pass through the midpoint is called a chord.**JEE Circles is a base to the entire syllabus of Coordinate Geometry in Mathematics. It is the easiest topic of coordinate geometry and with a bit of hard work, it becomes very easy to answer all the questions of this topic.**The preliminary knowledge of the concept of Straight Lines is required to study Circles. Circles introduce a lot of concepts in coordinate geometry. This is also important for complex numbers. Do this topic well. Get good command over locus problems. Practice lots of problems on various forms of equations of a circle. Give special attention to the parametric form.**Study the JEE circles with a good understanding of the diagrams and graphs. This will reduce the effort that is needed to solve any problem in this topic. This is so because having a clear picture of the problem statement in your mind is always helpful.**Determining your approach is as crucial as the process of actually studying. Keep twisting each question you solve. Make room for new possibilities and it will help you to understand it more deeply.**JEE Circles: Important books**You should first go through the basic concepts through the NCERT/R.D. Sharma and for practicing advanced level, you can try Coordinate Geometry by S.K Goyal (Arihant Publications) or S.L. Loney (Coordinate Geometry). The former is a more JEE oriented and contains much relevant stuff. This is the best book for JEE Circle/Coordinate Geometry, which is written in such a way that even a beginner can understand clearly. It has questions and examples from basic level to advanced level. The solutions to examples are presented in such a manner that you can easily understand. After reading concepts, shortcuts, and examples anyone can solve all the problems with little efforts.**JEE Circles: Important Subtopics**Equation of a circle in various forms, Equation of tangent and normal in point form, parametric form, and slope form.**Length of tangent from a point, equation of chord of contact, equation of chord with a given midpoint, equation of common tangent, finding radical axis, radical centre, etc.**Intersection of a circle with a straight line or circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line (family of circles).**Parametric equations of a circle, Number of tangents to a circle from a given point, and Orthogonal Circles(condition and Criteria).**JEE Circles: Some Basic Tips**Basically when a question is given, you should start with drawing a rough sketch of what the question is saying.**Try solving some simple examples (Level 1 — direct formula types) from any good material you have (R.D. Sharma or NCERT). It will definitely boost your self-confidence.**Jot down all relevant formulas on a sheet of paper. By the time you are up with level 1 problems, you must have learned most formulas. With the formulas and methods at your fingertips, you might consider going for Level 2 — Application based JEE circle problems (from S. K. Goyal).**Do not read the same concept from different books as it will lead to confusion. Always stick to one particular standard text for your concepts.**Do not practice even a single problem in coordinate geometry without drawing a figure. Mistaking the curves for one another is very common.**Many problems that involve numerical data come from the JEE circles. Try to avoid arithmetical errors in such problems.**If you don’t thoroughly understand a problem, drawing an approximate diagram according to the given scale can help you work out the radius, area, equations etc. easily. It might seem trivial, but it really works in many cases.**JEE Circles: Some Important points**The questions including straight line and circle that are asked in JEE can be solved using geometry and a few simple equations.**When it comes to questions on ‘Normal’ and you have no idea about the question, just put the center coordinate of given circle on the Normal line and you will surely get some hint (as Normal of a circle always passes through the center).**Whenever you are not getting any idea what to do, just write down the general equation of circle and start plugging in known parameters. You will probably get to a point where you’ll have one unknown parameter left. You can use your ordered pair to solve for that parameter.**Most problems on JEE circles can easily be solved using parametric form. Because it involves only one parameter (say θ).**JEE circles is an especially important topic because concepts of circles are used in complex number and area under the curve.**The formulas derived in this topic are general and they can be applied to ellipse, parabola, and hyperbola as well.**JEE Circles: Some Interesting Facts**Director Circle is the locus of all points from which two perpendicular tangents may be drawn to the circle. Questions are asked indirectly from this topic.**Orthogonality: Two circles cut each other orthogonally if the angle of intersection of the given circles at the point of intersection is a right angle. You must know condition and criteria for orthogonality.**Radical Axis & Radical Center: The radical axis of two circles is the locus of points at which tangents drawn to both circles have the same length. And the radical lines of three circles are concurrent at a point known as the radical center. (Important for Jee Main)*