Circle

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COORDINATE GE0METRY

(CIRCLE)

By:- Nishant Gupta For any help contact: 9953168795, 9268789880


Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85 Contact: 9953168795, 9268789880

BASIC

1. Standard form of equation of a circle :

Equation of a circle whose centre is (h, k} and radius r units is

(x – h)2 + (y - k)2 = r2

If centre is (0, 0), then equation of circle is x2+y2 = r2

2. General form of equation of a circle:

A second degree equation in two variables x & y represents a circle if

(i) Coefficients of x2 and y2 are equal.

(ii) Terms containing the product of xy is missing.

General equation of a circle is x2 + y2 + 2gx + 2fy + c = 0.

Centre is ( - g, -f ) =( - 2

1coeff of x , -

2

1 coeff. Of y )

Radius is cfg 22

Note : While finding centre & radius a circle, we must make sure that the coefficients of x2 & y2 are 1 each.

3. Parametric equations of a circle : Parametric equations of a circle whose centre is (h,k) and radius r is

x = h + r cos θ , y = k + r sin θ, ( θ is parameter)

4. Diametric form:

If A (x1,y1 ) & B( x2, y2 ) be diameter of a circle then its equation is

(x – x1) (x – x2) + (y – y1) (y – y2) = 0

5. Intercepts on axes by x2 + y2 + 2gx + 2fy + c = 0

(a) On X – axis cg2 2

(b) On Y – axis cf2 2

ADDITIONAL

CIRCLE



1. Tangent :- To find tangent at (x1, y1) change x2 & y2 to xx1 & yy1, x & y to 2

1(x + x1) and

2

1(y +

y1) resp. (It is denoted by T=0 )

2. A line touches a circle if ┴ distance from the centre is equal to radius

y = mx + c is tangent to x2 + y2 = r2 if c2 = r2 ( 1+m2 ) & point of contact is ( - 22 1

,1 m

c

m

mc

) where c =

21 m

3. Chord with given mid-point (x1, y1) is given by T = S1 y2 = 4ax yy1 – 2a(x + x1) = y2 – 4ax

S1 is obtained by putting (x1, y1) in curve.

4. Joint equation of tangents from (x1, y1) is SS1 = T2, From (2, 3) to x2 + y2 = 16 is (x2 + y2 – 16) (22 + 32 – 16) = (2x + 3y – 16)2

5. Angle between tangents from an external point 2 tan-1 1S

r

6. Chord of contact :It is line joining points of contact of tangents from any external point to a curve. Its equation is given by T=0

7. Polar/Pole : Polar of any curve w.r.t. to P(x1, y1) is locus of point of intersection of tangents at extremities of chords through (x1, y1) & the point P is known as POLE.

Tangent, Polar & Chord of Contact have same equation T = 0

Conjugate pts.:- Two pts are said to be so if polar of each passes through other point.

8. Length of tangents from P to any circle is(After changing to standard form) 1S

Length of direct common tangent of two circles 2212 rrd

Length of alternate common tangent of two circles 2212 rrd

Where r1 ,r2 be radii & d be distance between their centers

NOTE : direct common tangent of two circles divides line joining their centres externally in the ratio of the radii while alternate divides in same ratio internally

9. Common chord of two circles S1 = 0 & S2 = 0 is S1 – S2 = 0

Just subtract after changing to standard forms

10. No. of common tangents to two circle with C1 & C2 centres & radii r1 & r2 & if

(i) c1c2 > r1 + r2 then 4 common tangents

(ii) c1c2 = r1 + r2 then 3 common tangents

(iii) | r1 – r2 | < c1c2 < r1 + r2 2 common tangents

(iv) c1c2 = r1 – r2 only 1 common tangent

(v) c1c2 < | r1 – r2 | . No common tangent

11. Director circle :- It is locus of point of intersection of tangents to any circle [for x2+y2=a2 Director circle is x2+y2=2a2]

It is a concentric circle whose radius is 2 times radius of the original circle

12. Angle between two circles : In general angle between two curves is angle between tangents at their point of intersection



If r1 ,r2 be radii & d be distance between their centers then angle is given by Cosθ =21

222

21

rr2

drr

13. ORTHOGONALITY CONDITION x2 + y2 + 2gi x + 2f iy + ci= 0 where i = 1, 2

2 (g1 g2 + f1 f2 ) = c1 + c2

NOT IN SYLLABUS OF AIEEE

14. Radical axis :- Locus of a point which moves s.t. tangents from it to two circleS are of same length.

Equation S1 – S2 = 0 (same as common chord)

15. Radical centre of S1 = 0, S2 = 0 and S3 = 0. Find radical axes S1 – S2 = 0 & S2 – S3 = 0 Their point of intersection is radical centre.

16. Limiting point; Circle with radius zero. If (0, 0) is one limiting point then other is

2222 gf

fc,

gf

gc

17. Co-axial system :- If every pair of a family of circle have same radical axis.

We have may take its equation as x2 + y2 + 2gx + c = 0



1. 2x2 +2kxy + 2y2 + (k-4)x +6y –5 = 0

represents circle of radius

(a) 3√2 (b) 2√3

(c) 2√2 (d) N/T

2. Equation of the circle centred at (1, 2) and passing through the point of intersection of the lines x+ 2y= 3 and 3x +y =4.

(a) x2 + y2 -x-2y+1=0

(b) x2 + y2 - 2x-4y +4=0

(c ) x2 + y2 - 2x-4y -4=0

(c) N/T

3. If circle concentric with the circle x2 + y2 + 4x-2y + 4 = 0 and with three times its radius.

(a) x2 + y2 - 4x-2y -4=0

(b) x2 + y2 + 4x-2y -1=0

(c) x2 + y2 + 4x-2y- 4=0

(d) N/T

4. The equation of diameter of circle x2 + y2 -2x + 4 y = 0 which passes through origin is

(a) x +2y = 0 (b) x – 2y = 0

(c) 2x +y = 0 (d) 2x – y = 0

5. If 4x – 3y- 7 = 0 and 8x – 6y -39 = 0 are common tangents to a circle, then radius is

(a) 5/2 (b) 7/2

(c) 5/4 (d) 3/4

6. Equation of the circle having diameters 2x – 3y = 5 and 3x – 4y = 7 and radius 8 is

(a) x2 + y2 + 2x + 2y – 2 = 0

(b) x2 + y2 + 2x - 2y + 62 = 0

(c) x2 + y2 + 2x + 2y - 62 = 0

(d) none of these.

]

7. Number of tangents drawn from (- 5,2 ) to the circle x2 + y2 – 14x + 2y - 25 = 0 are

(a) 0 (b) 2

(c) 1 (d) N/T

8. Tangent to x2 + y2 + 4x - 4y + 4 = 0 making equal intercepts on axes ,is

(a) x+ y =2 (b) x+ y =4

(c) x+ y=2√2 (d) x+ y = 8

9. m = ? such that y = mx + 2 cuts x2 + y2 = 1 at distinct / coincident pts

(a) [ - , - 3 ] U [ 3 , ] (b) [ - 3 , 3 ]

(c) [ 3 , ] (d) None

10. x + y tanθ = cos θ touches x2 + y2 = 4 for θ

(a) π/4 (b) π /3

(c) π /2 (d) N/T

11. Two circles each of radius 5, have common tangent at (1, 1) whose eqn. is 3x+4y-7= 0. Then their centers are

(a) (3, 4) , ( -2,3) (b) (4, -3), (-2,5)

(c) ( 4, 5), (-2-3) (d) N/T

12. The length of the tangent drawn form any point on the circle x2 + y2 + 2gx + 2fy + = 0 to the circle x2 + y2 + 2gx + 2fy + = 0

(a) ( - ) (b) ( - )

(c) ( + ) (d) None.

13. If circle x2 + y2 – 6x – 4y + 9 = 0 bisects circumference of the circle x2 + y2 – 6y + k = 0, then k equals

(a) 2 (b)26

(c) 15 (d) 1

ASSIGNMENT CIRCLE



14. From a point on x2 + y 2 = a2 , two tangents are drawn to x2 + y 2 = a2 sin2 α then angle between them is

(a) α (b)α/2

(c) 2α (d) N/T

15. Locus of point of intersection of tangents to circle x = rcos θ , y = r sin θ at points whose parametric angles differ by π / 3

(a) x2 + y2 = r2 /3 (b) x2 + y2 = 4r2 /3

(c ) x2 + y2 = 2r2 /3 (d) N/T

16. C1 , C2 , C3 .. .. .. be a sequence of circles such that Cn+ 1 is director circle of Cn . If C1 has radius a then area bounded by Cn & Cn+ 1 is

(a) π 2n a2 (b) π 2n-1 a2

(c) π 2n-2 a2 (d) π 22n-2 a2

17. If radii of smallest & largest circles passing through ( √3 , √2 ) & touching x2 + y 2 – 2 √2 y – 2 = 0 are r 1 & r2 resp then AM of r1 & r 2 is

(a) 1 (b) √2

(c) √3 (d) 2

18. Equation of circle through intersection of x2 + y 2 + 2x = 0 & x – y = 0 & having minimum radius , is

(a) x2 + y 2 -1=0 (b) x2 + y 2 –x-y =0

(c) x2 + y 2 –2x-2y =0 (d) x2 + y 2 –x + y =0

19. Chord through ( 2 , 1) to x2 + y2 - 2 x – 2 y +1= 0 is bisected at ( a , 1/2 ) then a is

(a) 1/2 (b) 1

(c) 0 (d) N/T

20. If one of the circles x 2 + y2 + 2ax + c = 0 , x 2 + y2 + 2bx + c = 0 lies with in the other , then

(a) ab > 0 , c > 0 (b) ab > 0 , c < 0

(c) ab < 0 , c > 0 (d)N/T

21. A ray of light incidents at ( -3, -1 ) , gets reflected from tangent at ( 0 , -1 ) to x2 + y 2 = 1 . If reflected ray touches the circle , then equation of reflected ray is

(a) -3x + 4y = 5 (b) -2x +3y = 3

(c) 3x - 2y = 7 (d) N/T

22. Equation of circle through & cutting intercepts of length a & b on axes

(a) x2 + y2 + ax+ by = 0

(b) x2 + y2 – ax – by = 0

(c) x2 + y2 + bx+ ay = 0

(d) x2 + y2 – bx-ay = 0

23. The four distinct points (0, 0), (2, 0), (0 – 2) and (k, - 2) are concyclic if k =

(a) 5 (b) 6

(c) 7 (d) 8

24. aix + biy + ci = 0 (i = 1,2) cut axes in concyclic points then

(a) a1a2 = b1b2 (b) a1b1 = a2b2

(c) a1b2 = a2b1 (d) N/T

25. Number of integral values of k such that x2 + y2 + k x + ( 1-k) y + 5= 0 represents a circle whose radius cannot exceed 5 is

(a) 14 (b) 16

(c) 18 (d) 20

26. If A(1,4) &B are symmetrical about tangent to x2 + y2 - x + y = 0 at origin then B is

(a) ( 1, 2 ) (b) ( √2 , 1 )

(c) ( 4 , 1 ) (d) N/T

27. Tangents to x2 + y2 – 2gx – 2fy + f2 = 0 from origin are r if

(a) g2 + f 2 = 1 (b) g2 - f 2 = 1

(c) g2 - f 2 = 0 (d) N/T

28. If a line is drawn through point P(, ) to cut

circle x2 + y2 = a2 at A & B, then PA . PB =

(a) 2 + 2 (b)2+ 2 -a2

(c) a2 (d) N/T

29. Tangents to x2 + y2 – 2px – 2qy + q2 = 0 from origin are r if

(a) p2 = q2 (b) p2 + q2 = 1

(c) p = q/2 (d) q = 1/ 2

30. Locus of center of circle thro’ ( 1,2 ) & cutting x2 + y2 = 4 orthogonally is

(a) x2 + y2 – 3x – 8y + 1 = 0

(b) x2 + y2 – 6x - 2y = 7

(c) 2x + 4y – 9 = 0

(d) 2x + 4y – 1 = 0



31. Chord of contact of tangents from P to x2 + y

2 = a2 always touches x2 + y 2 – 2ax = 0 , locus of P is

(a) y 2 = a( a-2x) (b) x2 + y 2 = ( x- a) 2

(c) x 2 = a( a-2y) (d) N/T

32. The locus of the center of the circle x2 + y2 + 4x cos - 2 y sin - 10 = 0 is

(a) an ellipse (b) a circle

(c)a hyperbola (d) a parabola

33. Locus of poles of 222 ayx & so that

polar is always touching 222 byx is

(a) 4222 b)yx(a (b) 4222 a)yx(b

(c) 2222 bayx (d) None

34. Distances of centers of x2 + y2 – 2 ix = c2 ( i= 1,2,3 ) from origin are in GP then lengths of tangents from any pt. of x2 + y2 = c2 to these circles are in

(a) AP (b) GP

(c) HP (d) None

35. If chord of contact of tangents drawn from a

pt. on 222 ayx to 222 byx touches

222 cyx , then a, b, c are in

(a) A.P (b) G.P

(c) H.P (d) None

36. If abscissae & ordinates of points P & Q are roots of x2 + 2ax = b2 & x2 + 2px = q2 resp circle with P Q as diameter is

(a) x2 + y2 + 2ax + 2py – b2 – q2 = 0

(b) x2 + y2 –2ax –2py + b2 + q2 = 0

(c) x2 + y2 – 2ax–2py– b2 – q2 = 0 (d) None

37. Circles x2 + y2 + x + y = 0 & x2 + y2 + x – y = 0 intersect at an angle of

(a) /6 (b) /4

(c) /3 (d) N/T

38. Image of x2 + y2 – 6x + 8 = 0 & y = x is

(a) x2 + y2 – 6y + 8 = 0 (b) x2 + y2 – 6y = 8

(c) x2 + y2 + 6x + 8 = 0 (d) N/T

39. Greatest distance of point (10, 7) from circle x2 + y2 – 4x – 2y – 20 = 0 is

(a) 10 (b) 15

(c) 5 (d) N/T

40. Two circles, each of radius 5, have a common tangent at (1, 1) whose equation is 3x + 4y – 7 = 0. Then their centres are

(a) (4, -5), (-2, 3) (b) (4, -3), (-2, 5)

(c) (4, 5) , (-2, -3) (d) N/T

41. If 4yx 22 & 94yx 22 =0 have

two common tangent then is

(a)

8

13,

8

13 (b)

8

13 or

8

13

(c) 8/13 (d) None

42. Two circles x2 + y2 = 6 and x2 + y2 – 6x + 8 = 0 are given. Then circle through their point of intersection and the point (1, 1) is

(a) x2 + y2 – 6x + 4 = 0 (b) x2 + y2 – 3x + 1 = 0

(c) x2 + y2 – 4y + 2 = 0 (d) N/T

43. Let PQ & RS be tangents at extremities of diameter PR of circle of radius r . If PQ & RS intersect at X on circumference of the circle , then 2r is

(a) RS.PQ (b)2

RSPQ

(c) RSPQ

RS.PQ2

(d)

2

RSPQ 22

44. Two circles of radii a and b ( a > b ) touch each other externally. Then the radius of circle which touches both of them externally and also their direct common tangent is

(a) 2ba

ab

(b) √ab

(c) (a+b ) /2 (d) N/ T

45. If tangents to 1b

y

9

x2

22

through ( 1 ,2√3 )

are at right angles then b is

(a) 2 (b) 3

(c) 4 (d) 1



ANSWER (CIRCLE)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

d b c c c d a c a d c a c c b

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

b a d d c a b a a b c c b a c

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

ab a b b b a d a b c d b a a a

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