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.
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 .
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
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.
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.
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.
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
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.
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.
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
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
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
EXAMPLESFind the equation of the normal to the circlex2 + y2 – 2x – 4y + 3 = 0 at the point (2, 3). Ans. x – y + 1 = 0
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
LENGTH OF A TANGENT AND POWER OF A POINT
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
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.
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.
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.
SYSTEM OF CIRCLES
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.
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.
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.
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
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
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
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
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
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.
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
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.