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LONG ANSWER TYPE QUESTIONS:

1.If ( )2,0 ( )( ) ( )0,1 4,5 and 0,c are con cyclic then find c. 2.P.T (-2,8),(1,1),(-6,0),(-2,2) are con cyclic

3. Find direct common tangent equations for circles x2 + y2 + 22x − 4y−100 = 0

and x2 + y2 − 22x + 4y+100 = 0

4.Find the equation of the circle passing through the points (4,1) (6,5) and having the center on the line 4 16 0x y+ − =

5. Prove that the circles x 2+ y 2+6x +18y +26 =02 2 4 6 12 0x y x y+ − − − =

touch each other, find the point of contact o & equation of common tangent at the point of contact 6. Find the equation of the circle, whose radius is ‘5’ and which

touches the circle 2 2 2 4 20 0x y x y+ − − − = at (5,5)

7.S.T the line 5 12 4x y+ = touches the circle

2 2 6 4 12 0x y x y+ − + + = and find the point of contact

8. Show that 2 2 2 26 9 13 0, 2 16 0x y x y x y x y+ − − + = + − − = touch each other .Find the point of contact

and the equation of common tangent at their point of contact 9. Find the equation of the circle which cuts the following circles orthogonally

2 2 2 2 2 24 2 1 0,2( ) 8 6 3 0, 6 2 3 0x y x y x y x y x y x y+ + + + = + + + − = + + − − =

10 If the straight line 2 3 1x y+ = intersects the circle 2 2 4x y+ = at the points A and B then find the

equation of the circle having AB as diameter

11. (i)If 3x y+ = is the equation of the chord AB of circle 2 2 2 4 8 0x y x y+ − + − = ,find the equation of the

circle having AB as diameter (ii).Prove that common chord circles x 2+ y 2-6x -4y +9=0. x 2+ y 2-8x -6y +23= 0 is a diameter of

second circle 12. Equation of a parabola in standard form.

13. Evaluate 1

1+ sin x + cos xdx∫

14. Evaluate

2x +5

x2 − 2x +10∫ dx

15. (i)9cos sin4sin 5cos

x x dxx x

−+∫

(ii)2cos 3sin4cos 5sin

x x dxx x++∫ (iii)

2sin 3cos 43sin 4cos 5

x xx x+ ++ +∫

16.Evaluate ( ) 2

11 3 2

dxx x x+ + −∫

17.Evaluate ( ) 26 5 6 2x x xdx= + − +∫

18. 21x x x dx+ −∫

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19. Evaluate 23 8 3x x dx+ −∫

20. Evaluate Integration of 1 / 5- 2x2 + 4x SHORT ANSWER QUESTIONS

1. Show that the circles 2 2 2 0x y ax c+ + + = and

2 2 2 0x y by c+ + + = touch each other if 2 2

1 1 1a b c

+ =

2. Find the equation of the circle whose diameter is the common chord of the circles

2 2S x y 2x 3y 1 0≡ + + + + = and 2 2S' x y 4x 3y 2 0≡ + + + + =

3. If the length of the latus rectum is equal

152 and distance between

foci is 2 then find equation of ellipse in the standard form

4. Find the eccentricity and the length of the latus rectum of the 9x2 +16y2 − 36x + 32y− 92 = 0 5. Find the centre, eccentricity, foci, directrix and the length

of the Latus rectum of the hyperbola. 2 24x 9y 8x 32 0− − − =

6. Find the number of possible common tangents that exists for the following pairs of circles

2 2 2 26 6 14 0 , 2 4 4 0x y x y x y x y+ + + + = + − − − =

7. Find the internal center of similitude for the circles

2 2 6 2 1 0x y x y+ + − + = and

2 2 2 6 9 0x y x y+ − − + = 8. Find the external center of similitude for the circles

2 2 2 6 9 0x y x y+ − − + = and

2 2 4x y+ =

9. Find the direct common tangents of the circles 2 2 22 4 100 0x y x y+ + − − = and

2 2 22 4 100 0x y x y+ − + + = .

10. Find the transverse common tangents of the circle 2 2 4 10 28 0x y x y+ − − + =

and 2 2 4 6 4 0x y x y+ + − + =

11.. If the abscissae of points A,B are the roots of the equation 2 22 0x ax b+ − = ordinates of A,B are roots of 2 22 0y py q+ − = then find the equation of a circle for which AB is a diameter

12. Find the equation of circle passing through each of the following three points (3,4) ,(3,2), (1,4) 13, Find the equation of the circle passing through (0,0) and making intercept 6 units on x- axis and

intercept 4 units on Y- axis. 14. Find the equations of the tangents to the circle 2 2 4 6 12 0x y x y+ − + − = which are parallel to

8 0x y+ − = . 15. Show that 1 0x y+ + = touches the circle 2 2 3 7 14 0x y x y+ − + + = and find its point of contact 16. Find the chord of contact of (1,1) to the circle 2 2 9x y+ = 17 Find the value of k if 5 0x y+ − = and 2 8 0x ky+ − = are conjugate with respect to the circle

2 2 2 2 1 0x y x y+ − − − = 18, Find the value of k if kx +3y-1 =0 and 2x + y + 5=0 are conjugate with respect to the circle

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x2 +y2 -2x-4y-4=0 19 Show that tangent at (-1,2)of the circle x2 +y2 -4x-8y -7=0 touches the circle x 2+y2 +4x+6y =0 and find point of contact. 20. Find the pair of tangent drawn from (1,3) to the circle 2 2 2 4 11 0x y x y+ − + − = and also find the angle

between them 21. Find the length of chord intercepted by the circle x 2+ y2 - x +3y-22=0 on the line y =x-3 22. Find the equation of a parabola in the standard form

23. 1 ,xTan x dx x R− ∈∫

24 Find the equation of the circle which passes through the origin and intersects the circles below,

orthogonally

2 2 2 24 6 10 0, 12 6 0x y x y x y y+ − + + = + + + =

25. Find the equation of the circle which passes through the points (2,0), (0,2) and orthogonal to the

circle 2 22 2 5 6 4 0x y x y+ + − + =

26. Find the radical center of the following circles.

2 2 4 6 5 0x y x y+ − − + = ,

2 2 2 22 4 1 0, 6 2 0x y x y x y x y+ − − − = + − − =

27. Find the equation of the tangent and normal to the parabola 2 1 0y x y− + + = at (3,1)

28. Find the equation of the tangents to the ellipse 2 22 8x y+ = which are

i) parallel to 2 4 0x y− − = ii) perpendicular to 2 0x y+ + =

iii) which makes an angle 4π

with x- axis.

29. Find the equations of the tangents to the hyperbola 2 23 4 12x y− = which are

(i) parallel and (ii) perpendicular to the line y=x-7

30. Find the equation of normal to the hyperbola 2 23 144x y− = at the end of the latus rectum (first

quadrant ). VERY SHORT ANSWER TYPE QUESTIONS

1. Find circle concentric with x2 + y2 − 6x − 4y −12 = 0 and passing through(-2,14)

2. For circle x2 + y2 −10x −10y+ 25= 0 find polar equation drawn from (1,-2)

3. Find angle between circles x2 + y2 −12x − 6y+ 41= 0 , x

2 + y2 + 4x + 6y−59 = 0

4. Find the equation of axis and directrix of the parabola 2 6 2 5 0y y x+ − + =

.5. If the eccentricity of a hyperbola is

54 , then find the eccentricity of its conjugate hyperbola.

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6. Evaluate

( )( )

x

2 x

e 1 xdx

cos xe+

∫.

7. Find { }

21 cos { \ :1 cos2

x dx on I R n n Zx

π+ ⊂ ∈−∫

8. Find k if the following pair of circles are orthogonal 2 2 2 22 0, 2 8 0x y by k x y ax+ + − = + + + = 9. Find the angle between the circles given by the equations 2 2 2 212 6 41 0, 4 6 59 0x y x y x y x y+ − − + = + + + − = Show that the circles given by the following interrect each other orthogonally 10. Find the equation of the radical axis of the following circles ( )2 2 2 23 4 5 0,3 7 8 11 0x y x y x y x y+ − − + = + − + + =

11 Show that A(3,-1) lies on the circle 2 2 2 4 0x y x y+ − + = . Also find the other end of the diameter

through A. 12. Find the equation of the circle with center (2,3) and touching the line 3x-4y+1=0

13. Find the equation of tangent and normal at (3,2) is the circle 2 2 3 4 0x y x y+ − − − =

14 Find the chord of contact of (1,1) to the circle 2 2 9x y+ =

15 Find the polar of (1,-2) with respect to 2 2 10 10 25 0x y x y+ − − + =

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SHORT ANSWER TYPE QUESTIONS

1.# ( )( ) ( ){ }2

1\ : cos 0

cos

xx

x

e xdx on I R x R xe

xe+

⊂ ∈ =∫

Evaluate the following integrals

2. ( ) ( )2log

0,xdx on

x∞∫

Evaluate the following integrals. 3. ( )log 0, ,nx x dx on n∞∫ is a real number and 1n ≠ − .

4. 2cosx xdx on R∫

5. ( )2sec 2 \ 2 1 :4

x x dx on I R n n Zππ⎧ ⎫⊂ + ∈⎨ ⎬⎩ ⎭∫ .

6. 1 ,xTan x dx x R− ∈∫

7. ( ) ( )sin , , , , 0axe bx c dx a b c R b on R+ ∈ ≠∫

8. 2

sin2 cos

dθ θθ−∫

9. 2

cossin 4sin 5

x dxx x+ +∫

10. 2 22sin 3cosdxx x+∫

11. 12 3cos2

dxx−∫

13. ( )( )

2

21 2x dx

x x+ +∫

14. ( )( )2

31 1x dx

x x+

− +∫

15. ( )( )2

11 4

dxx x− +∫

16. ( )( )1 2

dxx x x+ +∫

17. ( ) ( )2

7 41 2x dx

x x−

− +∫

18. ( )( )( )

1x a x b x c− − −∫

19. ( )( )2

2 33 4x dx

x x+

+ +∫

20. 2

sin coscos 3cos 2

x x dxx x+ +∫

21. 2 3xx e dx−∫

22. Find ( )2

5,5

x dx onx

− ∞+∫

23. Evaluate 21

dxx x+ −∫

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24. Evaluate 5 4cosdx

x+∫

25. Evaluate ( )5 4

dxx x+ +∫

27. Evaluate 2 1dx

x x+ +∫

28. Evaluate { }1 sin \ 2 :1 cos

x xe dx on I R n n Zx

π−⎛ ⎞ ⊂ ∈⎜ ⎟−⎝ ⎠∫

29. Evaluate ( )1 1 , 1,11xTan dx onx

− − −+∫

30. 1x x dxonRe e−+∫

31. ( )

( )2sec \ 2 1 :

2sec tanx dxon I R n n z

x xπ⎧ ⎫⊂ + ∈⎨ ⎬

⎩ ⎭+∫

32. 2

cos3sin 4sin 5

x dxon Rx x− +∫

LONG ANSWER TYPE QUESTIONS