12th Standard Maths English Medium Two Dimensional Analytical Geometry-II Reduced Syllabus Important Questions 2021

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

Multiple Choice Questions

15 x 1 = 15
1. The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is

(a)

$\frac { 4 }{ 3 }$

(b)

$\frac { 4 }{ \sqrt { 3 } }$

(c)

$\frac { 2 }{ \sqrt { 3 } }$

(d)

$\frac { 3 }{ 2 }$

2. The circle x2+y2=4x+8y+5intersects the line3x−4y=m at two distinct points if

(a)

15< m < 65

(b)

35< m <85

(c)

−85<m < −35

(d)

−35<m <15

3. The area of quadrilateral formed with foci of the hyperbolas $\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\\$ and $\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } =-1$

(a)

4(a2+b2)

(b)

2(a2+b2)

(c)

a2 +b2

(d)

$\frac { 1 }{ 2 }$(a2+b2)

4. Tangents are drawn to the hyperbola  $\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } =1$ 1parallel to the straight line2x−y=1. One of the points of contact of tangents on the hyperbola is

(a)

$\frac { 9 }{ 2\sqrt { 2 } } ,\frac { -1 }{ \sqrt { 2 } }$

(b)

$\frac { -9 }{ 2\sqrt { 2 } } ,\frac { 1 }{ \sqrt { 2 } }$

(c)

$\frac { 9 }{ 2\sqrt { 2 } } ,\frac { 1 }{ \sqrt { 2 } }$

(d)

$\left( 3\sqrt { 3 } ,-2\sqrt { 2 } \right)$

5. An ellipse hasOB as semi minor axes, F and F′ its foci and the angle FBF′ is a right angle. Then the eccentricity of the ellipse is

(a)

$\frac { 1 }{ \sqrt { 2 } }$

(b)

$\frac { 1 }{ 2 }$

(c)

$\frac { 1 }{ 4 }$

(d)

$\frac { 1 }{ \sqrt { 3 } }$

6. y2 - 2x - 2y + 5 = 0 is a

(a)

circle

(b)

parabola

(c)

ellipse

(d)

hyperbola

7. If a parabolic reflector is 20 em in diameter and 5 em deep, then its focus is

(a)

(0,5)

(b)

(5,0)

(c)

(10,0)

(d)

(0, 10)

8. The equation 7x2- 6$\sqrt { 3 }$ xy + 13y2 - 4$\sqrt { 3 }$ x - 4y - 12 = 0 represents

(a)

parabola

(b)

ellipse

(c)

hyperbola

(d)

rectangular hyperbola

9. The distance between the foci of a hyperbola is 16 and e = $\sqrt { 2 }$ Its equation is

(a)

x2 - y2 = 32

(b)

y2 - x2 = 32

(c)

x2 - y2 = 16

(d)

y2 - x2 = 16

10. If the foci of the ellipse $\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ { b }^{ 2 } }$ = 1 and the hyperbola $\frac { { x }^{ 2 } }{ 144 } -\frac { { y }^{ 2 } }{ 81 } =\frac { 1 }{ 25 }$ coincide then b2 is

(a)

1

(b)

5

(c)

7

(d)

9

11. If e1,e2 are eccentricities of the ellipse $\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } }$ = 1 and the hyperbola $\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } }$ = 1 then

(a)

${ e }_{ 1 }^{ 2 }$ - ${ e }_{ 2 }^{ 2 }$ = 1

(b)

${ e }_{ 1 }^{ 2 }$  + ${ e }_{ 2 }^{ 2 }$ = 1

(c)

${ e }_{ 1 }^{ 2 }$ - ${ e }_{ 2 }^{ 2 }$ = 2

(d)

${ e }_{ 1 }^{ 2 }$ - ${ e }_{ 2 }^{ 2 }$=2

12. The equation of the director circle of the hyperbola $\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } }$ = 1 is ___________

(a)

x2 + y2 = a2+ b2

(b)

x2 +y2 = a2

(c)

x2+ y2 = b2

(d)

x2+y2 = a2- b2

13. The number of normals to the hyperbola $\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } }$ = 1 from an external point is

(a)

2

(b)

4

(c)

6

(d)

5

14. If t1 and t2 are the extremities of any focal chord of y2 = 4ax then t1tis ______________

(a)

-1

(b)

0

(c)

±1

(d)

$\frac12$

15. The locus of the foot of perpendicular from the focus on any tangent to y2 = 4ax is

(a)

x2 + y2 = a2 - b2

(b)

x2 + y2 = a2

(c)

x2 + y2 = a2 - b2

(d)

x = 0

1. 2 Marks

10 x 2 = 20
16. Find the general equation of a circle with centre(-3,-4) and radius 3 units.

17. If y=4x+c is a tangent to the circle x2+y2=9 , find c .

18. Find the equation of the circle with centre (2,-1) and passing through the point (3,6) in standard form.

19. Find centre and radius of the following circles.
x2+ (y+2)2 =0

20. Find centre and radius of the following circles.
2x2+2y2−6x+4y+2=0

21. Find the equation of tangent to the circle x2 +y2 + 2x - 3y - 8 = 0 at (2, 3).

22. If the line y = 3x + 1, touches the parabola y2 = 4ax, find the length of the latus rectum?

23. For the ellipse x2 + 3y2 = a2, find the length of major and minor axis.

24. Find the eccentricity of the hyperbola. with foci on the x-axis if the length of its conjugate axis is ${ \left( \frac { 3 }{ 4 } \right) }^{ th }$ of the length of its tranverse axis.

25. Find the equation of the hyperbola whose vertices are (0, ±7) and e = $\frac { 4 }{ 3 }$

1. 3 Marks

10 x 3 = 30
26. Find the equation of the circle described on the chord 3x+y+5= 0 of the circle x2+y2=16 as diameter.

27. Find the centre and radius of the circle3x2+(a+1)y2+6x−9y+a+4=0.

28. Find the equation of the circle with centre (2,3) and passing through the intersection of the lines 3x−2y−1=0and 4x+y−27=0.

29. A circle of area 9π square units has two of its diameters along the lines x+y=5 and x−y=1.
Find the equation of the circle.

30. If the equation3x2+(3−p)xy+qy2−2px =8pq represents a circle, find p and q . Also determine the centre and radius of the circle

31. Find the length of Latus rectum of the ellipse$\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1$

32. Find the equation of the hyperbola with vertices (0,±4) and foci(0,±6).

33. Find the equation of the ellipse in each of the cases given below:
(i) foci(±3 0),e =$\frac { 1 }{ 2 }$
(ii) foci (0,±4)and end points of major axis are(0,±5).
(iii) length of latus rectum 8, eccentricity =$\frac { 3 }{ 5 }$ and major axis on x -axis.
(iv) length of latus rectum 4 , distance between foci4 $\sqrt{ 2}$  and major axis as y - axis.

34. Find the vertex, focus, equation of directrix and length of the latus rectum of the following:
y2 = −8x

35. Find the circumference and area of the circle x2 +y2 - 2x + 5y + 7 = 0

1. 5 Marks

7 x 5 = 35
36.  A road bridge over an irrigation canal have two semi circular vents each with a span of 20m and the supporting pillars of width 2m. Use Fig.5.16 to write the equations that model the arches.

37. For the ellipse4x2+y2+24x−2y+21 = 0 , find the centre, vertices, and the foci. Also prove that the length of latus rectum is 2 .

38. Find the equations of the two tangents that can be drawn from (5,2) to the ellipse
2x2+7y2=14 .

39. Certain telescopes contain both parabolic mirror and a hyperbolic mirror. In the telescope shown in figure the parabola and hyperbola share focus F1 which is 14mabove the vertex of the parabola. The hyperbola’s second focus F2 is 2m above the parabola’s vertex. The vertex of the hyperbolic mirror is 1m below F1. Position a coordinate system with the origin at the centre of the hyperbola and with the foci on the y-axis. Then find the equation of the hyperbola.

40. Parabolic cable of a 60m portion of the roadbed of a suspension bridge are positioned as shown below. Vertical Cables are to be spaced every 6m along this portion of the roadbed. Calculate the lengths of first two of these vertical cables from the vertex.

41. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

42. A kho-kho player In a practice Ion while running realises that the sum of tne distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him of the distance between the poles is 6m.