9463138669|NDA Coaching Center In Jalandhar|ANAND CLASSES|NDA Coaching Near Me

1. Call – 9463138669 – ANAND CLASSES VISIT :https://anandclasses.co.in/

2. CIRCLES

3. INTRODUCTION

4. DEFINITION OF A CIRCLE P r . A circle is a set of points in the plane which is equidistant from a fixed point. C The fixed point is called as centre. The distance between the centre and any point on the Circle is called Radius.

5. If the radius is one unit, then the circle is called a unit circle. • If the radius is zero, then the circle is called a point circle.

6. EQUATION OF A CIRCLE P(x1, y1) r Theorem: The equation of the circle with centre C (h,k) and radius r is (x-h)2+(y-k)2=r2 C(h,k) Proof : Let P(x1,y1) be a point on the locus  CP=r P lies on the circle   (x1-h)2 + (y1-k)2 = r2  The locus of P is (x-h)2+(y-k)2 = r2 • The equation of a circle with centre at the origin and radius r is x2+y2 = r2

7. Theorem: If g2+f2-c0 then the equation x2+y2+2gx+2fy+c=0 represents a circle with centre (-g,-f ) and radius is Proof : Given equation is x2+y2+2gx+2fy+c=0 x2+2gx+y2+2fy =-c adding g2+f2 on both sides. x2+2gx+g2+y2+2fy+f2 = g2+f2-c (x+g)2+(y+f)2 = g2+f2-c

8. [x-(-g)]2+[y-(-f)]2 = It represents a circle Centre = (-g, -f) Radius = r =

9. NOTE [x-(-g)]2+[y-(-f)]2 = ...............(i) • If g2+f2-c>0, equation(i) represents a real circle with centre (-g,-f ). • If g2+f2-c=0, the equation(i) represents a circle whose centre is (-g,-f) and radius is zero. i.e. the circle coincides with the centre and so it represents a point (-g,-f) called as a point circle.

10. If g2+f2-c<0 radius of the circle is imaginary. In this case, there are no real points on the circle and so it is called a virtual circle or imaginary circle.

11. Theorem: The conditions that the equation ax2+2hxy+by2+2gx+2fy+c = 0 to represent a circle are i) a = b  0 ii) h = 0 iii) g2+f2-ac0 Proof : We know that equation of a circle is of the form x2+y2+2Gx+2Fy+C=0(1) If ax2+2hxy+by2+2gx+2fy+c=0 (2) represents a circle Comparing (1) and (2) we get and h = 0

12.  a = b, h = 0 G2+F2-C  0  g2+f2-ac  0

13. In every equation of a circle • coefficient of x2 = coefficient of y2and coefficient of xy =0 • If ax2+ay2+2gx+2fy+c = 0 represents a circle, then its centre = and its • radius =

14. S=0 • If the circle S=0and the line L=0 intersect, then the equation of the circle passing through the points of intersection of the circle and the line is S+L=0 where  is a parameter L=0 • If a line passing through a point P(x1,y1) intersects the circle S=0 at the points A and B then PA.PB=S11. B A P

15. DEFINITIONS SECANT LINE Let A, B be two points on the circle, then 1. A secant is a line that intersects the circle at two points. B A

16. 2. A segment of line joining the points ‘A’ and ‘B’ is called chord. 3. AB is called the length of the chord of the circle. A B Chord on the circle C A B P AB=length of chord

17. y Theorem: The angle in a semicircle is a right angle. P(x1,y1) O x' x Proof : B(-r,0) A(r,0) Let O=(0,0) be the centre and A(r,0), B(-r,0) be the ends of the diameter AB of the circle x2+y2=r2. y' Let P(x1,y1) be a point on the circle. Equation of a circle with centre (0,0) and radius r is x2+y2=r2  x12 + y12 = r2

18.  Slope of is (m1) =  Slope of is (m2) =  m1m2 = = = (∵ x12 – r2 = -y12) = -1  m1m2 = -1  to  AB = 900

19. Theorem:- The equation of a circle having the segment joining A(x1,y1), B(x2,y2) as diameter is (x-x1) (x-x2)+(y-y1)(y-y2) = 0 A (x1,y1) B (x2,y2) Proof : Given that A(x1,y1), B(x2,y2) are the ends of the diameter Let P(x, y) be any point on the circle. APB = 900 Slope of  Slope of = -1

20.  (y-y1) (y-y2) = - (x-x1) (x-x2)  (x-x1) (x-x2) + (y-y1) (y-y2)=0 Equation of a circle having as diameter is x2+y2-(x1+x2)x-(y1+y2)y+x1x2+y1y2 = 0

21. Mcqs 1. Centre of the circle x2+y2+2gx+2fy+c=0 is… 1) (g,f) 2) (-g,f) 3) (-g,-f) 4) (g,-f) KEY : 3

22. 2. Radius of the circle x2+y2+2gx+2fy+c=0 is… 1) 2) 3) 4) KEY : 2

23. 3. Centre of the circle ax2+ay2+2gx+2fy+c=0 is… 1) 2) 3) 4) KEY : 3

24. 4. Centre of the circle x2+y2+4x+6y-12=0 is… 1) 2) 3) 4) KEY : 2

25. 5. Radius of the circle x2+y2+4x+6y-12=0 is… 1)25 2)5 3)1 4) 2 KEY : 2

26. NOTATIONS,CONCENTRIC CIRCLES & CONCYCLIC POINTS

27. NOTATIONS S = x2+y2+2gx+2fy+c S1 = xx1+yy1+g(x+x1)+f(y+y1)+c S11 = x12+ y12+ 2gx1+2fy1+c S12 = x1x2+y1y2+g(x1+x2)+f(y1+y2)+c

28. POWER OF A POINT Suppose S=0 is the equation of circle with centre ‘C’. Let P(x1,y1) be any point in the plane then CP2-r2 is defined as power of ‘p’ w.r.t to S = 0 i) If ‘P’ exterior of circle then power is positive ii) If ‘P’ on the circle then power is zero ii) If ‘P’ interior of circle then power is negative.

29. POSITION OF A POINT p c Let S = 0 be a circle in a plane and P(x1,y1) be any point in the same plane Then (i) ‘P’ is the interior of circle S11 < 0 Since P lies inside the circle, CP<r CP2 <r2 (x1+g)2+(y1+f)2 < r2

30. x12+ y12+2gx1+2fy1+g2+f2 < g2+f2-C x12+ y12+2gx1+2fy1+c < 0 S11 < 0 ii) P lies on the circle cp2-r2= 0 S11 = 0

31. iii) P lies outside the circle cp2-r2> 0 S11 > 0

32. CONCYCLIC POINTS Theorem: If the lines a1x+b1y+c1 = 0, a2x+b2y+c2 = 0 cuts the coordinate axes in concyclic points then prove that a1a2 = b1b2 Proof : Given that L1 a1x+b1y+c1 = 0 L2 a2x+b2y+c2 = 0

33. Meets the coordinate axes at ‘P’ and ‘Q’, L2=0 meets at R and S Y S Then coordinates of P,Q,R,S are Q R O X P

34. Since P,Q,R,S are concyclic OP.OR = OQ.OS  a1a2 = b1b2

35. If the lines a1x+b1y+c1 = 0, a2x+b2y+c2 = 0, meet the coordinate axes in four distinct concyclic points, then the equation of the circle passing through these concyclic points is (a1x+b1y+c1) (a2x+b2y+c2)- (a1b2+a2b1)xy = 0

36. Y Theorem: The equation of the circumcircle of the triangle formed by the line ax+by+c= 0 with coordinate axes is ab(x2+y2)+c(bx+ay) = 0 B Proof : X O A The line ax+by+c=0 cuts the coordinate axes at

37.  is a diameter of circle, so the equation of circle is  x2+y2+c = 0  ab(x2+y2)+c(bx+ay) = 0

38. CONCENTRIC CIRCLES • Two circles are said to be concentric if their centres are same • The equation of any circle concentric with the circle x2+y2+2gx+2fy+c=0 is x2+y2+2gx+2fy+=0, where  is any real constant.

39. Mcqs 1. The position of the point (1,2) with respect to the circle x2+y2+6x+8y-96=0 is… 1) outside the circle 2) inside the circle 3) on the circle 4) none KEY : 2

40. 2. The power of the point (2,4) with respect to the circle x2+y2-4x-6y-12=0 is… 1) -48 2) 48 3) 24 4) -24 KEY : 4

41. Thank you…