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The circle is an important concept in geometry that you must have studied in Class 9, Class 10 and Class 11. Many questions in competitive exams and engineering entrance tests include questions from the Circles chapter. This is why askIITians experts have compiled revision notes for Circles to help you revise all the concepts, theorems, and formulae related to Circles in no time. These notes include all the important topics based on the engineering entrance exams syllabus. With these notes, you can revise concepts related to circles for JEE, NEET, CBSE board exams and more.
The main topics covered in these online revision notes for Circles are the equation of the circle, equation of the chord, basic properties of circles, length of the chord of a circle, properties of chords of a circle, tangents to a circle, properties of tangents to a circle, the radical axis of circles, and equations of the family of circles. You can also check Maths Notes for Class 9, 10, 11, 12 separately on our website if you want to practice every concept thoroughly.
Equal chords of a circle subtend equal angles at the centre
The below table describes the equations of a circle according to changes in radii and centres:
The point P(x1, y1) lies outside, on, or inside a circle.
S ≡ x2 + y2 + 2gx + 2fy + c = 0, according as S1 ≡ x12 + y12 + 2gx1 + 2fy1 + c > = or < 0.
xx1 + yy1 + g(x + x1) + f(y + y1) = x12 + y12 + 2gx1 + 2fy1 i.e. T = S1
Length of the chord = 2r sin (c/2), where ‘c’ is the central angle and ‘r’ is the radius
If a circle has two secants QR and ST, then
Hence, in the below figure, if OB is perpendicular to PQ, then PA = AQ.
Suppose there are two circles with centres at C1 and C2 and radii as r1and r2 and P is the point of intersection of direct common tangents as shown in the figure below:
C1A1, C2A2 are perpendiculars from C1 and C2 to one of the tangents. The point ‘P’ divides C1C2 externally in the ratio r1: r2. So, to find the direct common tangent we just need to find the point P which divides the line joining the centre externally in the ratio of radii.
The equation of direct common tangent is SS1 = T2 where S is the equation of one circle.
S ≡ x2 + y2 + 2gx + 2fy + c and
S’ ≡ x2 + y2 + 2g’x + 2fy’ + c’
Then equation of radical axis of two circles S = 0 and S’ = 0 is given by S = S’
2(g – g’)x + 2(f – f’)y + (c – c’) = 0
2x(g – g’) + 2y(f – f’) + c – c’ = 0
If S’ = 0 and S = 0 touch each other, then S – S’ = 0 is the equation of the common tangent to the two circles at the point of contact.
2gg1+ 2ff1 = c + c
Circles is an important topic for JEE Main and Advanced. You can expect 1-2 questions based on circles in these examinations. So you must practice its concepts in school as well as in coaching classes. This will help you score higher percentile in the entrance tests.
You study all the basic concepts related to circles in Class 10 such as tangents to a circle, sector of a circle, the segment of a circle, etc. This knowledge is further applied in learning conic sections in Class 11.
askIITians provides Class 6, 7, 8, 9, 10, 11, 12 notes for Maths in a chapter-wise format. You can refer to our online revision notes and practice all Maths concepts on your own.
Enrol in online Maths coaching for CBSE, JEE, NEET and other popular school boards and learn in live, interactive classes from our experts. We also provide dedicated study notes for all the students like chapter notes, NCERT solutions, video lectures, mind maps, flashcards, study planners and much more. All these resources can help you master the Maths subject for competitive exams.
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Solved Examples on Circle Illustration 1:...