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# Circles

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### Circles

1. 1. CIRCLES
2. 2. DEFINITION The locus of a point, which moves in a plane, such that its distance from a fixed point in the plane is always constant, is called a circle.
3. 3. EQUATIONS OF CIRCLES Central Form : The equations of the circle with centre (h, k) and radius b ‘r’ is (x h)2 (y k)2 r 2. Standard Form : The equation of the circle with centre (0, 0) and radius ‘r’ is x2 y2 r2 .
4. 4.  General Form : The equation x2 y2 2gx 2fy c 0, where g, f and c are constants, represents a circle.i. Its centre is ( g, f ).ii. Its radius is g 2 f 2 c2, where g2 f2 – c 0.iii. Length of intercept made by the circle on x-axis is 2 g 2 c , where g2 c 0 and on y-axis is 2 f 2 c , where f2 – c 0. The general equation of second degree in x, y : ax2 2hxy by2 2gx 2fy c 0 represents a circle if and only if : (i) a b ii) h 0 (iii) g2 f2 – ac 0
5. 5.  Diameter Form : The equation of the circle when end- points of a diameter are A (x1, y1) and B (x2, y2) is : (x x1)(x x2) + (y y1)(y y2) 0 Equation of the circle concentric with x2 y2 2gx 2fy c 0 is : x2 y2 2gx 2fy k 0 , where k R.
6. 6. EXAMPLES Find the equation of the circle passing through the point of intersection of the lines x + 3y = 0 and 2x – 7y = 0 and whose center is the point of intersection of the lines x + y + 1 = 0 and x – 2y + 4 = 0. Ans. x2 + y2 + 4x – 2y = 0 Find the equation of the circle whose center is (1, 2) and which passes through the point (4, 6) Ans. x2 + y2 – 2x – 4y – 20 = 0 Find the equation of a circle whose radius is 6 and the center is at the origin. Ans. x2 + y2 = 36.
7. 7. EXAMPLES
8. 8. PARAMETRIC EQUATIONS If ‘ ’ is a parameter, where 0 2 , then :a) the equations x r cos ,y r sin are parametric equations of the circle x2 y2 r 2.b) the equations x h r cos ,y h r sin are parametric equations of the circle (x h)2 (y k)2 0.
9. 9. EXAMPLES Find the parametric equations of circle x2 + y2 – 6x + 4y – 12 = 0 Ans. x = 3 + 5 cos , y = –2 + 5 sin Find the cartesian equations of the curve x = –2 + 3 cos , y = 3 + 3 sin Ans. (x + 2)2 + (y – 3)2 = 9
10. 10. POSITION OF A POINT WITH RESPECT TO A CIRCLEThe point (x1, y1) is inside, on or outside the circle S x2 + y2 + 2gx + 2fy + c = 0.according as S1 x1² + y1² + 2gx1 + 2fy1 + c < , = or > 0.Note : The greatest & the least distance of a point A from acircle with center C & radius r is AC + r &AC - r respectively.
11. 11. EXAMPLESHow are the points (0, 1) (3, 1) and (1, 3) situated withrespect to the circle x2 + y2 – 2x – 4y + 3 = 0?Ans. (0, 1) lies on the circle ; (3, 1) lies outside thecircle ; (1, 3) lies inside the circle.
12. 12. CONDITION OF TANGENCY The condition that the st. line y = mx + c may touch the circle x2 + y2 = r2 is c r l m2 The equations of the tangents, in the slope-form, are y mx r 1 m2
13. 13. EXAMPLES For what value of , does the line 3x + 4y = touch the circle x2 + y2 = 10x. Ans. 40, –10 Find the equation of the tangents to the circle x2 + y2 – 2x – 4y – 4 = 0 which are (i) parallel, (ii) perpendicular to the line 3x – 4y – 1 = 0 Ans. (i) 3x – 4y + 20 = 0 and 3x – 4y – 10 = 0 (ii) 4x + 3y + 5 = 0 and 4x + 3y – 25 = 0
14. 14. TANGENT AND NORMAL The equation of the tangent at (x1,y1) to the circle :  x2 y2 r2 is xx1 yy1 r2 .  x2 y2 2gx 2fy c 0 is xx1 yy1 2g(x x1) 2f(y y1) c 0. The equation of the normal at (x1, y1) to the :  Circle x2 y2 r2 is xy1 x1y 0 i.e. x y x1 y1  Circle x2 y2 2gx 2fy c 0 is y y1 x x1 y1 f x1 g Length of the tangent from (x1, y1) to the circle x2 + y2 + 2gx + 2fy + c = 0 is : 2 2 x1 y1 2gx1 2fy1 c
15. 15. EXAMPLESFind the equation of the normal to the circlex2 + y2 – 2x – 4y + 3 = 0 at the point (2, 3). Ans. x – y + 1 = 0
16. 16. PAIR OF TANGENTS FROM A POINTThe equation of a pair of tangents drawn from the point A(x1, y1) to the circle x2 + y2 + 2gx + 2fy + c = 0 is SS1 = T².Where S x2 + y2 + 2gx + 2fy + c ; S1 x1² + y1² + 2gx1 + 2fy1 + c T xx1 + yy1 + g(x + x1) + f(y + y1) + c.Example : Find the equation of the tangents through(7, 1) to the circle x2 + y2 = 25. Ans. 12x2 – 12y2 + 7xy – 175x – 25y + 625 = 0
17. 17. LENGTH OF A TANGENT AND POWER OF A POINT
18. 18. EXAMPLES Find the area of the quadrilateral formed by a pair of tangents from the point (4, 5) to the circle x2 + y2 – 4x – 2y – 11 = 0 and a pair of its radii. Ans. 8 sq. units If the length of the tangent from a point (f, g) to the circle x2 + y2 = 4 be four times the length of the tangent from it to the circle x2 + y2 = 4x, show that 15f2 + 15g2 – 64f + 4 =0
19. 19. CHORD IN TERMS OF MIDDLE POINT The equation of the chord of the circle SL x2 y2 2gx 2fy c 0 , whose mid-point is (x1, y1), is : xx1 yy1 g(x x1) f(y y1) c x12 y12 2gx1 2fy1 c 0 i.e. T = S1.
20. 20. EXAMPLES
21. 21. DIRECTOR CIRCLE
22. 22. CHORD OF CONTACT If from a point P, PT and PT’ are two tangents to the circle, then TT’ is the chord of contact. The equation of the chord of contact of tangents drawn from P (x1, y1) to the circle1) x2 + y2 = r2 is xx1 + yy1 = r2.2) x2 + y2 + 2gx + 2fy + c = 0 is xx1 + yy1 + g (x + x1) + f (y + y1) + c = 0.
23. 23. EXAMPLES
24. 24. POLE AND POLAR If P be any point. Let any secant through P meet the circle at Q and R. The tangents at Q and R meet at S. Then the locus of S is called the polar of P and P is called the pole of the locus of S. The equation of the polar of P (x1, y1) w.r.t. the circle:1) x2 + y2 = r2 is xx1 + yy1 = r22) x2 + y2 + 2gx + 2fy + c = 0 is xx1 + yy1 + g (x + x1) + f (y + y1) + c = 0.
25. 25. EXAMPLES
26. 26. SYSTEM OF CIRCLES
27. 27. FAMILY OF CIRCLES Through line and circle : The equation of the family of circles passing through the points of intersection of the line L≡ lx + my + n = 0 and the circle S≡ x2 + y2 + 2gx + 2fy + c = 0 is (x2 + y2 + 2gx + 2fy + c) + λ(lx + my + n) = 0 , where λ is the parameter.
28. 28.  Through two circles : The equation of the family of circle passing through the points of intersection of two circles : S1≡ x2 + y2 + 2g1x + 2f1y + c1 = 0 and S2≡ x2 + y2 + 2g2x + 2f2y + c2 = 0 is : S1 + λS2 = 0 or (x2 + y2 + 2g1x + 2f1y + c1) + λ(x2 + y2 + 2g2x + 2f2y + c2) = 0 where λ is the parameter.
29. 29.  Through two points : The equation of the family of circles passing through two points (x1, y1) and (x2, y2) is : x1 y1 1( x x1 )( x x2 ) ( y y1 )( y y2 ) x2 y2 1 0 x3 y3 1where λ is the parameter.
30. 30.  Touching a line : The equation of the family of circles touching the line at (x1, y1) is: 2 2 x x1 y y1 ax by c 0 where λ is the parameter. Touching both axes: The equation of the family of circles touching both the axes is : 2 2 2 x y  2ay a 0
31. 31. ANGLE BETWEEN TWO CIRCLES The angle of intersection between two circles whose radii are r1 and r2 and d, the distance between their centre is 1 r12 r22 d 2 cos 2r2 r2
32. 32. ORTHOGONAL CIRCLES If two circles cut orthogonally (i.e. cut at right angles), then the square of the distance between their centres is equal to the sum of the squares of their radii. Condition of Orthogonality : The condition that the two circles : S1≡ x2 + y2 + 2g1x + 2f1y + c1 = 0 and S2≡ x2 + y2 + 2g2x + 2f2y + c2 = 0 may cut orthogonally is : 2g1g2 + 2f1f2 = c1 + c2
33. 33. EXAMPLES For what value of k the circles x2 + y2 + 5x + 3y + 7 = 0 and x2 + y2 – 8x + 6y + k = 0 cut orthogonally. Ans. – 18 Find the equation to the circle which passes through the origin and has its center on the line x + y + 4 = 0 and cuts the circle x2 + y2 – 4x + 2y + 4 = 0 orthogonally. Ans. 3x2 + 3y2 + 4x + 20y = 0
34. 34. COMMON CHORD Let S1≡ x2 + y2 + 2g1x + 2f1y + c1 = 0 and S2≡ x2 + y2 + 2g2x + 2f2y + c2 = 0 be the intersecting circles. Then the equation of the common chord is : 2 g1 g 2 x 2 f1 f 2 c1 c2 0
35. 35. COMMON TANGENTS TO THE CIRCLES Let S1 = 0 and S2 = 0 be two circles with radii r1 and r2 and d, the distance between their centres.1. When r1 - r2 > d, there is no common tangent. Here one circle is completely within the other.2. When r1 - r2 = d, there is one common tangent. Here circles touch each other internally.3. When r1 + r2 > d or r1 - r2 < d, there are two common tangents. Here the circles intersect each other in two distinct points.4. When r1 + r2 = d, there are three common tangents.5. When r1 + r2 < d, there are four common tangents.
36. 36. KEY POINTS1. Fixed point is the centre and constant distance is the radius.2. For circle : Coeff. of x2 = Coeff. of y2 Coeff. of xy = 0.3. Any point on the circle (x-h)2 + (y-k)2 = r2 is (h + r cos θ, k + r sin θ) where 0 ≤ θ ≥ 2 .4. Condition of Tangency is also given as c2 = r2 (l + m2).5. Equation of the tangent in slope-form is y mx r 1 m2
37. 37. 6. Normal at any point on the circle always passes through its centre.7. Equation of the chord in terms of mid-point is : T= S1.8. Equation of tangents from (x1, y1) to S = 0 is SS1 = T2.9. Equation of chord of contact is same as equation of the tangent.10. Equation of the polar is same as the equation of the tangent.