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Final step Parabola By Abhijit Kumar Jha

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SET�I

1. If y2 � 2x � 2y + 5 = 0 is(A) a circle with centre (1, 1) (B) a parabola with vertex (1, 2)

(C) a parabola with directrix x = 2

3(D) a parabola with directrix x =

2

1

2. The focus of the parabola x2 � 8x + 2y + 7 = 0 is

(A)

2

1,0 (B)

2

9,4 (C) (4, 4) (D)

2

9,4

3. Equation of a common tangent to the circle, x2 + y2 = 50 and the parabola, y2 = 40x can be(A) x + y - 10 = 0 (B) x - y + 10 = 0(C) x � y � 10 = 0 (D) none of these

4. The line y = mx + c touches the parabola x2 = 4ay if

(A) c = � am (B) c = m

a (C) c = � am2 (D) 2m

a

5. The length of the latus rectum of the parabola 4y2 + 2x � 20y + 17 = 0 is

(A) 3 (B) 6 (C) 2

1(D) 9

6. In a parabola semi-latus rectum is the harmonic mean of the(A) segments of a focal chord (B) segments of the directrix(C) segments of a chord (D) none of these

7. If (t2 , 2t) is one end of a focal chord of the parabola, y2 = 4x then the length of the focal chordwill be :

(A) tt

12

(B) tt

1t

t2

2

1

(C) tt

1t

t2

2

1

(D) none of these

8. The equation of the parabola with its vertex at (1, 1) and focus at (3, 1) is(A) (x � 3)2 = 8(y � 1) (B) (y �1)2 = 8(x � 1)(C) (y � 1)2 = 8(x � 3) (D) (x � 1)2 = 8(y � 1)

9. Equation of the tangent at (� 4, � 4) on x2 = � 4y is(A) 2x � y + 4 = 0 (B) 2x + y � 4 = 0(C) 2x � y � 12 = 0 (D) 2x + y + 4 = 0

10. The locus of the midpoint of the line segment joining the focus to a moving point on the parabolay2 = 4ax is another parabola with directrix

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(A) x = � a (B) x = a (C) x = 0 (D) 2

ax

11. The point of intersection of the tangents to the parabola at the points t1 and t

2 is

(A) (a t1 t2 , a(t

1 + t

2) ) (B) (2 a t

1 t2 , a(t

1 + t

2) )

(C) (2a t1 t2 , 2a(t

1 + t

2) ) (D) none of these

12. The tangent drawn at any point P to the parabola y2 = 4ax meets the directrix at the point K. Then theangle which KP subtends at the focus is(A) 300 (B) 450 (C) 600 (D) 900

13. The equation of the parabola whose focus is the point (0, 0) and the tangent at the vertex isx � y + 1 = 0 is(A) x2 + y2 � 2xy � 4x + 4y � 4 = 0 (B) x2 + y2 � 2xy + 4x � 4y � 4 = 0(C) x2 + y2 + 2xy � 4x + 4y � 4 = 0 (D) x2 + y2 + 2xy � 4x � 4y + 4 = 0

14. If (x1, y

1), (x

2, y

2), (x

3, y

3) be three points on the parabola y2 = 4ax, the normals at which meet in a

point, then(A) y

1 + y

2 + y

3 = 0 (B) x

1 + x

2 + x

3 = 0

(C) y1 + y

2 + y

3 = 2a (D) x

1 + x

2 + x

3 = 4a

15. The normal at a point on y2 = 4x passes through (5, 0). There are three such normals one of which isthe axis. The feet of the three normals from a triangle whose centroid is

(A) (2, 0) (B) (0, 2) (C)

2

1,

2

1(D) (5, 0)

16. The locus of the point of intersection of the perpendicular tangents to the parabola x2 = 4ay is(A) y = a (B) y = �a (C) x = a (D) x = �a

17. The normal at the point P(ap2 , 2ap) meets the parabola y2 = 4ax again at Q(aq2, 2aq) such that thelines joining the origin to P and Q are at right angle: Then(A) p2 = 2 (B) q2 = 2 (C) p = 2q (D) q = 2p

18. If 2x + y + k = 0 is a normal to the parabola y2 = �8x, then the value of k is(A) �16 (B) �8 (C) �24 (D) 24

19. The angle between the tangents drawn from the origin to the parabola y2 = 4a(x�a) is(A) 90° (B) 45° (C) 60° (D) tan�12

20. The parametric coordinates of any point on the parabola y2 = x can be(A) (sec2 , sec ) (B) (sin2 , sin ) (C) (cos2 , cos ) (D) none of these




SET�II1. If the normals at points t

1 and t

2 meet on the parabola, then

(A) t1t2 = �1 (B) t

2 = � t

1 �

1t

2(C) t

1t2 = 2 (D) none of these

2. Which one of the following equations represented parametrically, represents equation to a parabolicprofile ?

(A) x = 3 cos t ; y = 4 sin t (B) x2 2 = 2 cos t ; y = 4 cos2 t

2

(C) x = tan t ; y = sec t (D) x = 1 sin t ; y = sin t

2 + cos

t

2

3. The condition that the two tangents to the parabola y2 = 4ax become normal to the circlex2 + y2 � 2ax � 2by + c = 0 is given by(A) a2 > 4b2 (B) b2 > 2a2 (C) a2 > 2b2 (D) b2 > 4a2

4. The locus of the middle points of focal chords of a parabola is(A) y2 = 2a(x � a) (B) y2 = 2a (x + a) (C) x2 = 2a (y � a) (D) x2 > 2a(y + a)

5. The normal chord of a parabola y2 = 4ax at (x1, x

1) subtends a right angle at the

(A) focus (B) vertex(C) end of the latusrectum (D) none of these

6. If the chord y = mx + c subtends a right angle at the vertex of the parabola y2 = 4ax, then the value ofc is(A) � 4am (B) 4 am (C) � 2am (D) 2 am

7. If P(�3, 2) is one end of the focal chord PQ of the parabola y2 + 4x + 4y = 0, then the slope of thenormal at Q is

(A) 2

1� (B) 2 (C) 2

1(D) �2

8. The angle between tangents to the parabola y2 = 4ax at the points where it intersects with the linex�y�a = 0 is

(A) 3

(B)

4

(C)

6

(D)

2

9. If the lines (y � b) = m1(x + a) and (y � b) = m

2(x + a) are the tangents of y2 = 4ax then

(A) m1 + m

2 = 0 (B) m

1m

2 = 1 (C) m

1m

2 = � 1 (D) m

1 + m

2 = 1

10. The parabola y2 = 4x and the circle (x � 6)2 + y2 = r2 will have no common tangent if �r� is equal to(A) r > 20 (B) r < 20 (C) r > 18 (D) r ( 20 , 28

11. Parabolas y2 = 4a(x � c1) and x2 = 4a(y � c

2), where c

1 and c

2 are variable, are such that they touch

each other. Locus of their point of contact is(A) xy = 2a2 (B) xy = 4a2 (C) xy = a2 (D) none of these

12. Minimum distance between the curve y2 = 4x and x2 + y2 � 12x + 31 = 0 is equal to




(A) 21 (B) 26 � 5 (C) 21 � 5 (D) 28 � 5

13. The locus of the foot of the perpendicular from the focus upon a tangent to the parabolay2 = 4ax is(A) the directrix (B) tangent at the vertex(C) x = a (D) none of these

14. If (xr, y

r); r = 1, 2, 3, 4 be the points of intersection of the parabola y2 = 4ax and the circle

x2 + y2 + 2gx + 2fy + c = 0, then(A) y

1 + y

2 + y

3 + y

4 = 0 (B) y

1 + y

2 � y

3 � y

4 = 0

(C) y1 � y

2 + y

3 � y

4 = 0 (D) y

1 � y

2 � y

3 + y

4 = 0

15. The length of a focal chord of the parabola y2 = 4ax making an angle with the axis of the parabolais(A) 4a cosec2 (B) 4a sec2 (C) a cosec2 (D) none of these

16. Three normals to the parabola y2 = x are drawn through a point (C, 0) then

(A) C = 4

1(B) C =

2

1(C) C >

2

1(D) none of these

17. The triangle PQR of area 'A' is inscribed in the parabola y2 = 4ax such that the vertex P lies at thevertex of the parabola and the base QR is a focal chord . The modulus of the difference of theordinates of the points Q and R is :

(A) A

a2(B)

A

a(C)

2A

a(D)

4A

a

18. The tangent and normal at the point P(at2, 2at) to the parabola y2 = 4ax meet the x-axis in T and Grespectively, then the angle at which the tangent at P to the parabola is inclined to the tangent at P tothe parabola is inclined to the tangent at P to the circle through P, T, G is(A) tan-1(t2) (B) cot-1(t2) (C) tan-1(t) (D) cot-1(t)

19. From an external point P, pair of tangent lines are drawn to the parabola, y2 = 4x . If 1 &

2 are the

inclinations of these tangents with the axis of x such that, 1 +

2 =

4

, then the locus of P is :

(A) x - y + 1 = 0 (B) x + y - 1 = 0 (C) x - y - 1 = 0 (D) x + y + 1 = 0

20. Length of the chord of contact of the pair of tangents drawn from on the point (x1 , y

1) the parabola,

y2 = 4ax is :

(A) 1

a y ax y a1

21 1

2 24 4 (B) 1

2a y ax y a12

1 12 24 4

(C) 1

a y ax y a1

21 1

2 24 4 (D) 1

a y ax1

214 y a1

2 24




SET�III

1. Equation x2 � 2x � 2y + 5 = 0 represents(A) a circle with centre (1, 1) (B) a parabola with vertex (1, 2)(C) a parabola with directrix y = 5/2 (D) a parabola with directrix y = �1/3

2. The normals to the parabola y2 = 4ax from the point (5a, 2a) are(A) y = x � 3a (B) y = �2x + 12a (C) y = �3x + 33a (D) y = x + 3a

3. The equation of the lines joining the vertex of the parabola y2 = 6x to the points on it whoseabscissa is 24, is(A) y ± 2x = 0 (B) 2y ± x = 0 (C) x ± 2y = 0 (D) 2x ± y = 0

4. The equation of the tangent to the parabola y2 = 9x which goes through the point (4, 10) is(A) x + 4y + 1 = 0 (B) 9x + 4y + 4 = 0 (C) x � 4y + 36 = 0 (D) 9x � 4y + 4 = 0

5. Consider the equation of a parabola y2 + 4ax = 0, where a > 0. Which of the following is false ?(A) tangent at the vertex is x = 0 (B) directrix of the parabola is x = 0(C) vertex of the parabola is at the origin (D) focus of the parabola is at (a, 0)

Question based on write-upNormally, the various propositions you study, e.g. equation of tangent, normal, chord, focalchord, formula for focal distance etc, are derived for the parabola y2 = 4ax. However, all theresults with slight transformation are valid for any parabola. Suppose we represent the equationof parabola y2 � 4ax = 0 by S (x, y, a) = 0 and any equation derived for this parabola byP(x, y, a) = 0. Now, if the given parabola is y2 = � 4ax, y2 + 4ax = 0 we can write, ifS(x, y, � a) = 0, so the corresponding equation of P will be P(x, y, � a) = 0. Similarly forx2 = 4ay can be written as S(x, y, a) and corresponding transformation is P(x, y, a)(i.e. interchange x and y).

6. The focal distance of the point (x, y) on the parabola x2 � 8x + 16y = 0 is(A) | y � 4 | (B) | y � 5 | (C) | y � 2 | (D) | x � 4 |

7. Normals are drawn from the point (7, 14) to the parabola x2 � 8x � 16y = 0. The slopes of thesenormals are

(A) 1 (B) �2 (C) 2

3(D)

2

3

8. The coordinates of the feet of the normals obtained in previous problem are(A) (0, 8) (B) (4, 3) (C) (16, 8) (D) (�8, 4)

9. The points on the axis of the parabola x2 + 2x + 4y + 13 = 0 from the where three distinctnormals can be drawn are given by

(A) )2,1(k),k,1( (B) (�1, 6)(C) )5,(k),k,1( (D) (�1, 1)

10. The line psinycosx touches the parabola x2 + 4a(y + a) = 0, if




(A) a = p sec (B) acos2 = p sin(C) a2cos + p2 sin = 0 (D) a tan = p secThree normals can be drawn from a point (x

1,y

1) to the parabola y2 = 4ax. The points where

these normals meet the parabola are called the feet of the normals or conormal points. The sumof the slopes of the normals is zero and the sum of the ordinates of the feet of the normals is alsozero.

m1 + m

2 + m

3 = 0 y

1 + y

2 + y

3 = - 2a(m

1 + m

2 + m

3) = 0

m1 m

2 + m

2 m

3 + m

3 m

1 =

a

xa2 1

m1 m

2 m

3 =

a

y1 , where m1 , m

2 , m

3 are slopes and y

1 , y

2 , y

3 are ordinates.

11. The locus of the point of intersection of the three normals to the parabola y2 = 4ax, two of whichare inclined at right angles to each other is(A) y{y2 + (3a + x)a}= 0 (B) y{y2 + (3a � x)a}= 0(C) y{y2 � (3a � x)a}= 0 (D) none of these

12. The normals at three points P, Q, R of the parabola y2 = 4ax meet in (h, k). The centroid oftriangle PQR lies on(A) x = 0 (B) y = 0(C) x = � a (D) none of these

13. The number of distinct normals that can be drawn from

4

1,

4

11to the parabola y2 = 4x is

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

14. Three normals to the parabola y2 = x are drawn through a point (C, 0), then

(A) C = 4

1(B) C =

2

1(C) C >

2

1(D) none of these

15. If the normals from any point to the parabola x2 = 4y cuts the line y = 2 in points whose abscissaare in AP, then the slopes of the tangents at the three conormal points are in(A) G.P (B) A.P.(C) H.P. (D) none of these

16. The locus of the points such that two of the three normals from them to the parabolay2 = 4ax coincide is :(A) 27ay2 + 4 (x + 2a)3 = 0 (B) 27ay2 + 4 (x � 2a)3 = 0(C) 27ay2 = 4 (x � 2a)3 (D) none of these

17. Assertion : Solpe of tangents drawn from (4, 10) to parabola y2 = 9x are 4

9,

4

1.

Reason : Every parabola is symmetric about its directrix.(A) both Assertion and Reason are true and Assertion is the correct

explanation of �Reason�




(B) both Assertion and Reason are true and Assertion is not the correctexplanation of �Reason�

(C) Assertion is true but Reason is false(D) Assertion is false but Reason is true

18. True/False :

(i) The equation of the parabola whose focus is at the origin is y2 = 4a(x + a).

(ii) The locus of the mid�points of the chords of the parabola y2 = 4ax which pass throughthe vertex, is the parabola y2 = 2ax

(iii) The focus of the parabola x2 + 8y = 0 is at (0, 2)

(iv) The line y = mx + c is a tangent to the parabola y2 = 4a(x + a), then m

amac

(v) The points x = 2at, y = at2 are lies on the parabola x2 = 4ay.

19. Fill in the blanks :

(i) The equation of the parabola whose focus is the point (2, 3) and directrix is the linex � 4y + 3 = 0 is ...........and the length of its latus rectum is..............

(ii) For the parabola y2 + 4x � 6y + 13 = 0, the vertex is ........, focus is.........directrix is.........L.R. is.........

(iii) The length of the latus rectum of the parabola x2 � 4x � 8y + 12 = 0 is .................(iv) The focus of the parabola y = 2x2 + x is .................(v) The vertex of the parabola (y � 2)2 = 16(x � 1) is .................

20. Match the columnColumn I Column II

(a) The latus rectum of the parabola y2 = 5x + 4y + 1 is (P) x = 2

3

(b) Two perpendicular tangents to y2 = 4ax always intersect on the line (Q) 0

(c) The equation of the directrix of the parabola y2 + 4y + 4x + 2 = 0 is (R) 4

23

(d) The number of tangent(s) to the parabola y2 = 8x through (2, 1) is (S) x = � a

(e) If two different tangents of y2 = 4ax are the normals to x2 = 4bythen | b | is less than (T) 5

(f) Minimum distance between the curves y2 = x � 1 and x2 = y � 1 is (U) 22

1




LEVEL�I ANSWER

1. y = 2x2 + 3x + 4, LR = 2 unit 4.

3

4,

3

4,

3

1,

12

1

5. 5x + y = 135a 8. 3a4 unit

9. y = -4x + 72, y = 3x - 33 10. y2 = 4a(x - 8a)

LEVEL�II

2. )am4,am4(,m

a4,

m

a4 22

SET�I

1. C 2. C 3. B 4. C 5. C6. A 7. A 8. C 9. A 10. C11. A 12. D 13. C 14. A 15. A16. B 17. A 18. D 19. A 20. D

SET�II

1. C 2. B 3. D 4. A 5. A6. A 7. A 8. D 9. C 10. B11. B 12. C 13. B 14. A 15. A16. C 17. C 18. C 19. C 20. C

SET�III

1. BC 2. AB 3. BC 4. CD 5. BD6. B 7. C 8. C 9. C 10. B11. B 12. B 13. C 14. C 15. C16. C 17. C18. (i) T (ii) T (iii) F (iv) T (v) T

19. (i) 16x2 + y2 + 8xy � 74x � 78y + 212, 17

14(ii) (�1, 3), (�2, 3), x = 0, �4

(iii) 8 (iv)

0,

4

1(v) (1, 2)

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


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