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Final Step (ellipse) BY Abhijit Kumar jha

HTTP://AKJ259.WORDPRESS.COM

SET�I

1. If the locus of the point of intersection of perpendicular tangents to the ellipse x

a

y

b

2

2

2

21 is a

circle with centre at (0, 0), then the radius of the circle would be(A) a + b (B) ab

(C) b/a (D) ( )a b2 2

2. There are exactly two points on the ellipse x

a

y

b

2

2

2

21 whose distance from the center of the ellipse

are equal and equal to a b2 22

2

. Eccentricity of this ellipse is equal to

(A) 3

2(B)

1

3

(C) 1

2(D)

2

3

3. If the line y = mx + c, interests the ellipse x

a

y

b

2

2

2

21 , at points whose eccentric angles differ by

/ 3 , then(A) 3(a2 m2 + b2) = 4c2 (B) 3(a2 + b2 m2) = 4c2

(C) a2m2 + b2 = 4c2 (D) a2 + b2 m2 = 4c2

4. Consider an ellipse with major and minor axes of length 10 and 8 units respectively. The radius oflargest circle that can be inscribed in this ellipse, it is given that centre of this circle is one focus of theellipse, is equal to(A) 4 units (B) 2 units(C) 6 units (D) none of these

5. Eccentricity of the ellipse 5x2 + 6xy + 5y2 = 8 is

(A) 1

2(B)

3

2

(C) 2

3(D)

1

3

6. The tangent at the point '' on the ellipse x

a

2

2 + y

b

2

2 = 1 meets the auxiliary circle in two

points which subtends a right angle at the centre, then the eccentricity 'e' of the ellipse isgiven by the equation(A) e2 (1 + cos2 ) = 1 (B) e2 . (cosec2 1) = 1(C) e2 (1 + sin2 ) = 1 (D) e2 (1 + tan2 ) = 1

7. S and T are the foci of an ellipse and B is an end of the minor axis. If STB is equilateral,then e is

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(A) 1/4 (B) 1/3(C) 1/2 (D) none of these

8. A ladder 12 units long slides in a vertical plane with its ends in contact with a verticalwall and a horizontal floor along x-axis. The locus of a point on the ladder 4 units from itsfoot has the equation :

(A) x2

4 + y2 = 1 (B)

x y2 2

16 64 = 1

(C) x y2 2

64 16 = 1 (D) x2 +

y2

4 = 1

9. Eccentric angle of a point on the ellipse x2 + 3y2 = 6 at a distance 2 units from the centre of theellipse is(A) 2 3 / (B) / 3(C) 3 4 / (D) none of these

10. The point of intersection of the tangents at the point P on the ellipse x

a

y

b

2

2

2

2 = 1 and its

corresponding point Q on the auxiliary circle meet on the line :(A) x = a/e (B) x = 0(C) y = 0 (D) none of these

11. If and are eccentric angles of the ends of a focal chord of the ellipse x

a

y

b

2

2

2

21 , then

tan tan 2 2

is equal to

(A) 1

1

e

e(B)

e

e

1

1

(C) e

e

1

1(D)

e

e

1

3

12. The distances from the foci of P(a, b) on the ellipse x y2 2

9 251 are

(A) 45

4 b (B) 5

4

5 a

(C) 54

5 b (D) none of these

13. If tan 1

. tan 2 =

a

b

2

2 then the chord joining two points 1 &

2 on the ellipse 1

b

y

a

x2

2

2

2




will subtend a right angle at(A) focus (B) centre(C) end of the major axis (D) end of the minor axis

14. The equations of the common tangents to the ellipse, x2 + 4y2 = 8 & the parabolay2 = 4x are(A) x � 3 = ± 2y (B) x � 4 = ± 2y(C) x + 4 = ± 2y (D) none of these

15. If O is the centre, OA the semimajor axis and S the focus of an ellipse, the eccentric angle of anypoint P is(A) POS (B) PSA (C) PAS (D) none of these

16. If A and B are two fixed points and P is a variable point such that PA + PB = 4, the locus of P is(A) a parabola (B) an ellipse (C) a hyperbola (D) none of these

17. If P( ) and Q

2FHGIKJ are two points one the ellipse

x

a

y

b

2

2

2

21 , locus of mid-point of PQ is

(A) x2

a

y

b2

2

2

1

2 (B)

x2

a

y

b2

2

24

(C) x2

a

y

b2

2

22 (D) none of these

18. The length of the chord of the ellipse x y2 2

25 161 where mid-point is

1

2

2

3,FHGIKJ

(A) 1

10(B)

8161

10

(C) 8061

10(D) none of these

19. The sum of the square of perpendiculars on any tangent to the ellipse x

a

y

b

2

2

2

21 from two point

on the minor axis, each at a distance are from the centre, is(A) 2a2 (B) 2b2 (C) a2 + b2 (D) a2 - b2

20. If latus rectum of the ellipse x y2 2 2 2 1tan sec is 1/2 then ( )0 is equal to

(A) / 12 (B) / 6 (C) / 8 (D) none of these




SET�II

1. The length of the major axis of the ellipse (5x � 10)2 + (5y + 15)2 = 4

)7y4x3( 2is

(A) 10 (B) 3

20(C)

7

20(D) 4

2. The tangent and normal to the ellipse x2 + 4y2 = 4 at a point P( ) on it meets the major axes in Q and

R respectively. If QR = 2, then cos is equal to

(A) 5

4(B)

3

2(C)

3

1(D) none of these

3. The ellipse 1b

y

a

x2

2

2

2

and the straight line y = mx + c intersect in real points only if

(A) a2 m2 < c2 � b2 (B) a2 m2 > c2 � b2

(C) a2 m2 c2 � b2 (D) c b

4. The foci of the ellipse 25 (x + 1)2 + 9 (y + 2)2 = 225 are at(A) (�1, 2) and (�1, �6) (B) (�2, 1) and (�2, 6)(C) (�1, �2) and (�2, �1) (D) (�1, �2) and (�1, �6).

5. The parametric representation of a point on the ellipse whose foci are (�1, 0) and (7, 0) and eccen-tricity 1/2 is

(A) (3 + 8cos , 4 3 sin ) (B) (8cos , 4 3 sin )

(C) (3 + 4 3 cos , 8sin ) (D) none of these

6. The equation 2a

y

a6

x 22

= 1, will represent an ellipse if

(A) a (1, 3) (B) a(1, 6)(C) a (� , 2) (6, ) (D) a (2, 6) ~ {4}

7. Tangents are drawn to the ellipse x2 + 2y2 = 4 from any arbitrary point on the line x + y = 4, thecorresponding chord of contact will always pass through a fixed point, whose coordinates are

(A)

2

1,1 (B)

1,2

1(C)

2

1,1 (D)

1,

2

1

8. The line y = x � 1 touches the ellipse 3x2 + 4y2 = 12, at




(A)

2

1,

2

1(B) (3, 2) (C) (�1, �2) (D) None of these

9. The normal drawn to the ellipse 1b

y

a

x2

2

2

2

at the extremity of the latus rectum passes through the

extremity of the minor axis. Eccentricity of this ellipse is equal to

(A) 2

15 (B)

2

15 (C)

2

13 (D)

2

13

10. The line 5x � 3y = 28 is a normal to the ellipse 19

y

25

x 22

. If � � be the eccentric angle of the foot

of this normal, then � � is equal to

(A) 6

(B)

3

(C)

4

(D) None of these

11. Tangent drawn to the ellipse 1b

y

a

x2

2

2

2

at point �P� meets the coordinate axes at points A and B

respectively. Locus of mid-point of segment AB is

(A) 2b

y

a

x2

2

2

2

(B) 2y

b

x

a2

2

2

2

(C) 4y

b

x

a2

2

2

2

(D) 4b

y

a

x2

2

2

2

12. Tangents PA and PB are drawn to the ellipse 19

y

16

x 22

from the point P(0, 5). Area of triangle

PAB is equal to

(A) 5

16 sq. units (B)

25

256 sq. units (C)

5

32 sq. units (D)

25

1024 sq. units

13. Tangents are drawn to the ellipse 19

y

36

x 22

from any point on the parabola y2 = 4x. The

corresponding chord of contact will touch a parabola, whose equation is(A) y2 + 4x = 0 (B) y2 = 4x (C) y2 + 9x = 0 (D) y2 = 9x

14. The normal at a variable point P on an ellipse x

a

y

b

2

2

2

2 = 1 of eccentricity e meets the axes of the

ellipse in Q and R then the locus of the mid-point of QR is a conic with an eccentricity e suchthat(A) e is independent of e (B) e = 1(C) e = e (D) e = 1/e




15. An ellipse is such that the length of the latus rectum is equal to the sum of the lengths of its semiprincipal axes. Then(A) Ellipse bulges to a circle(B) Ellipse becomes a line segment between the two foci(C) Ellipse becomes a parabola(D) none of these

16. If the line 3x + 4y = 7 touches the ellipse 3x2 + 4y2 = 1 then, the point of contact is

(A) 1

7

1

7,

(B)

1

3

1

3,

(C) 1

7

1

7,

(D) none of these

17. A common tangent to 9x2 + 16y2 = 144 ; y2 x + 4 = 0 & x2 + y2 12x + 32 = 0 is(A) y = 3 (B) x = 4(C) x = 4 (D) y = 3

18. If F1 & F

2 are the feet of the perpendiculars from the foci S

1 & S

2 of an ellipse x y2 2

5 3 = 1 on the

tangent at any point P on the ellipse, then (S1F

1) . (S

2F

2) is equal to

(A) 2 (B) 3(C) 4 (D) 5

19. The area of the rectangle formed by the perpendiculars from the centre of the standard ellipse tothe tangent and normal at its point whose eccentric angle is /4 is

(A) a b ab

a b

2 2

2 2

(B)

a b ab

a b

2 2

2 2

(C)

a b

ab a b

2 2

2 2

(D)

a b

a b ab

2 2

2 2

20. If & are the eccentric angles of the extremities of a focal chord of an standard ellipse,then the eccentricity of the ellipse is :

(A) cos cos

cos( )

(B) sin sin

sin ( )

(C) cos cos

cos( )

(D) sin sin

sin ( )




SET�III

Multiple choice questions with one or more than one correct choice

1. If x - 2y + 4 = 0 is a common tangent to y2 = 4x & x y

b

2 2

24 = 1, then

(A) b = 3 (B) x + 2y + 4 = 0 (C) x + 2y � 4 = 0 (D) b = 3

2. If a number of ellipse be described having the same major axis 2a but a variable minoraxis then the tangents at the ends of their latusrectum pass through fixed points. Then the fixedpoints is/are(A) (0, a) (B) (a, a) (C) (0, - a) (D) (0, 0)

3. Eccentric angle of a point on the ellipse x2 + 3y2 = 6 at a distance 2 units from the centre of theellipse is

(A) 4

(B)

3

(C)

4

3(D)

3

2

4. For the ellipse 3x2 + 4y2 � 6x + 8y � 5 = 0

(A) centre is (1, �1) (B) eccentricity is 2

1

(C) foci are (3, �1) and (�1, �1) (D) all of these are true

5. If pair of tangents are drawn to the ellipse 16

x2

+ 9

y2

= 1 from a point P so that the tangents are

at right angles to each other then the possible co-ordinates of the point P is/are

(A) )7,23( (B) (5, 0) (C) (3, 4) (D) )5,52(

Read the following passage and answer the questions :

Consider the ellipse )ba(1b

y

a

x2

2

2

2

and circle x2 + y 2 = r2. Now any tangent of ellipse will

be 222 bmamxy and any tangent of circle will be 2m1rmxy .

6. The range of �r� for which 4 distinct common tangents are possible(A) [b, a] (B) (b, a) (C) (b, a] (D) [b, a)

7. The equation of common tangent in 4th quadrant will be

(A) 2 2 2 2

2 2 2 2

r b a by x r

a r a r

(B) 2 2 2 2

2 2 2 2

r b a by x r

a r a r




(C) 2 2 2 2

2 2 2 2

r b a by x r

a r a r

(D) 2 2 2 2

2 2 2 2

r b a by x r

a r a r

8. Area of quadrilateral formed by all the common tangent will be

(A) 2 2 2

2 2 2 2

2r (a b )

(a r ) (r b )

(B)

2 2 2

2 2 2 2

2r (a b )

(a b ) (a r )

(C) 2 2 2

2 2 2 2

4r (a b )

(a r ) (r b )

(D)

2 2 2

2 2 2 2

r (a b )

(a r ) (r b )

Read the passage given below and answer the questions :

Suppose that an ellipse and circle are respectively given by the equation

1b

y

a

x2

2

2

2

...........(i)

and x2 + y2 + 2gx + 2fy + c = 0 ..........(ii)

The equation 0)cfy2gx2yx(1b

y

a

x 222

2

2

2

..........(iii)

Represents a curve which passes through the common points of the ellipse (i) and the circle (ii).We can choose so that the equation (iii) represents a pair of straight lines. In general we get

three value of indicating three pair of straight lines can be through the points. Also when (iii)

represents a pair of straight lines they are parallel to the lines 0)yx(b

y

a

x 222

2

2

2

, which

represents a pair of lines equally inclined to axes (the term containing xy is absent). Hence twostraight lines through the points of intersection of an ellipse and any circle make equal angles withthe axes. Above description can be applied identically for a hyperbola and a circle.

9. The radius of the circle passing through the points of intersection of ellipse 1b

y

a

x2

2

2

2

and

x2 � y2 = 0 is

(A) 22 ba

ab

(B) 22 ba

ab2

(C) 22

22

ba

ba

(D) 22

22

ba

ba

10. If ,,, be eccentric angles of the four concyclic points of the ellipse 1b

y

a

x2

2

2

2

, then

is equal to




(A) 2

)1n2(

(B) )1n2(

(C) n2 (D) n

11. Let the eccentric angles of three points P, Q and R on the ellipse 1b

y

a

x2

2

2

2

are

2

, and

. A circle through P, Q and R cuts the ellipse again at S, then the eccentric angle of S is

(A) 3 (B)

32

3(C)

3

2(D)

3

2

12. Suppose two lines are drawn through the common points of intersection of hyperbola 2 2

2 2

x y1

a b

and circle x2 + y2 + 2gx + 2fy + c = 0. If these lines are inclined at angle and to x�axis then,

(A) (B) 2

(C) (D)

a

btan2 1

13. The number of pair of straight line formed by points of intersection of rectangular hyperbolax2 � y2 = 1 and circle x2 + y2 � 4x � 5 = 0 is(A) 0 (B) 1 (C) 3 (D) 2

Read the passage given below and answer the questions :

If PPC be a diameter of the ellipse and the diameter DDC be drawn parallel to the tangents at

P and P , then PPC be parallel to the tangents at D and D . Two such diameters are known as�conjugate diameters�.

Condition that the lines y = mx, y = xm should lie along conjugate diameters of the ellipse

1b

y

a

x2

2

2

2

is 2

2

a

bmm . If = an odd multiple of

2

.

14. Tangents at the extremities of conjugate diameters of the ellipse 1b

y

a

x2

2

2

2

intersect on the

ellipse




(A) 1b

y

a

x2

2

2

2

(B) 2 2

2 2

x y2

a b (C)

2 2

2 2

x y

a b = 3 (D) none of these

15. The locus of the intersection of normals at the extremities of conjugate diameters of the ellipse2 2

2 2

x y

a b = 1 is

(A) 2(a2 x2 + b2y2)3 = (a2 � b2)2 . (a2 x2 � b2y2)2

(B) 2(a2 x2 � b2y2)3 = (a2 � b2)2 . (a2 x2 + b2y2)2

(C) 2(a2 x2 � b2y2)3 = (a2 + b2)2 . (a2 x2 + b2y2)2

(D) none of these

16. If 2 2

2 2

x y

a b = 1 be an ellipse referred to two conjugate diameters as axes the lines y = mx ,

y = xm will be conjugate diameters if

(A) 2

2

b

amm (B) 2

2

a

bmm

(C) 2

2

a

bmm (D) none of these

17. CP and CDR conjugate semidiameters of an ellipse and the tangent at P meets any other pairs ofconjugate diameters in T and T , then

(A) 2CD2TP.TP (B) 2CDTP.TP

(C) 2CD

1TP.TP (D) none of these

18. True or False(i) Given the base of a triangle and sum of its sides then the locus of the centre of its in circle is an ellipse.

(ii) If a tangent of slope m at a point of the ellipse 1b

y

a

x2

2

2

2

passes through (2a, 0) and if �e� denotes

the eccentricity of the ellipse, then 3m2 + e2 = 1.

(iii) If the latus rectum of an ellipse is equal to half the minor axis, then its eccentricity is equal to 4

3. D

(iv) A line of fixed length (a + b) moves so that its ends are always on two fixed perpendicularstraight lines. The locus of the point which divided this line into portions of lengths a & b is anellipse

(v) An ellipse slides between two perpendicular straight lines . Then the locus of its centre is anellipse




19. Fill in the blanks :

(i) The eccentricity of an ellipse whose latus rectum equals half its major axis is ______ .

(ii) The equations of the common tangents to the ellipse, x2 + 4y2 = 8 & the parabola y2 = 4x are______ & ______ .

(iii) The equation of the ellipse with its centre at (1, 2), focus at (6, 2) and passing through the point(4, 6) is ______ .

(iv) P & Q are corresponding points on the ellipse x2

16 +

y2

9 = 1, and the auxiliary circle respectively..

The normal at P to the ellipse meets CQ in R where C is centre of the ellipse. Then l (CR) = ____

(v) The sum of the squares of the reciprocals of two perpendicular diameters of the ellipse,5x2 + 4y2 = 1 is equal to ______ .

20. Match the columnColumn I Column II

(a) The length of the semi latus rectum of an ellipse is onethird of its major axis, then its eccentricity would be (P) y-axis

(b) The point from which the tangents to the ellipse 5x2 + 4y2 = 20are perpendicular, is (Q) (1, 1)

(c) An ellipse has OB as a semi minor axis . F, F are its fociand the angle FBF is a right angle. Then the eccentricity

of the ellipse is (R) 1 2 2,d i

(d) The centre of the ellipse 16

)2yx( 2 + 1

9

)yx( 2

is (S)3

1

(e) The equation of normal at the point (0, 3) of the

ellipse 9x2 + 5y2 = 45 is (T)1

2




SET�I1. D 2. C 3. A 4. D 5. B6. C 7. A 8. C 9. C 10. C11. B 12. C 13. B 14. C 15. D16. B 17. A 18. D 19. A 20. A

SET�II

1. B 2. B 3. C 4. A 5. A6. D 7. A 8. D 9. A 10. C11. C 12. B 13. C 14. C 15. A16. D 17. C 18. B 19. A 20. D

SET�III

1. AB 2. AC 3. AC 4. AB 5. ABCD6. B 7. C 8. A 9. B 10. C11. C 12. C 13. C 14. B 15. A16. B 17. B18. (i) T (ii) T (iii) F (iv) T (v) F

19. (i) e = 1

2(ii) x + 4 = ± 2y (iii) ( ) ( )x y

1

45

2

201

2 2

(iv) 7 units (v) 9/4

20. a-S, b-R, c-T, d-Q, e-P


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