TECHNO ACE : AEP 2024

TO
ASSIGNMENT SHEET 2025

Class: XI : Subject: Mathematics

Assignment Number: TACE-AS: 11/24-25/SM: 2/PA-2/M-10

TOPIC (Circle, Parabola, Ellipse and Hyperbola)

OBJECTIVE
LEVEL : 1

1. The equation x2 + y2 – 6x + 8y + 25 = 0 represents

A a point B a pair of straight lines

C a circle with non zero radius D parabola

2. The locus of a point which moves such that the sum of the squares of its distance from three vertices of a
triangle is constant is

A a straight line B a circle C an ellipse D a parabola

3. The number of points with integral coordinates that are interior to the circle x2 + y2 = 16, is

A 43 B 49 C 45 D 51

4. If an equilateral triangle is inscribed in the circle x2 + y2 = a2, the length of its each side is,

3
A 2a B a C 3a D None of these
2
5. The radius of the smallest circle passing through the points (1, 10) and (2, 2) is

1 1 1
A B C D None of these
4 2 2
6. Consider a family of circles which are passing through the point (–1, 1) and are tangent to x – axis. If (h, k)
are the coordinates of the centre of the circles, then the set of values of k is given by the interval
1 1 1 1 1
A – K B K C 0 <k< D K
2 2 2 2 2
7. 2 2 2 2
The length of the transversal common tangent to the circles x + y = 1 and (x – t) + y = 1 is 21 , then t =
A  2 B  5 C  3 D None of these

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8. The equation of the smallest circle passing through the intersection of the line x + y = 1 and the circle x2 + y2
= 9, is

A x2 + y2 + x + y – 8 = 0 B x2 + y2 – x – y – 8 = 0 C x2 + y2 – x + y – 8 = 0 D None of these

9. The length of latus rectum of the parabola x = ax2 + by + c is, a > 0

a a 1 1
A B C D
4 3 a 4a
10. If M is the foot of the perpendicular from a point P on a parabola to its directrix and SPM is an equilateral
triangle, where S is the focus, then SP is equal to

A a B 2a C 3a D 4a

11. The length of the talus rectum of the parabola 2 {(x – a)2 + (y – a)2} = (x + y)2 is

A 2a B 2 2a C 4a D 2a

12. TThe vertex of the parabola

x2 + y2 – 2xy – 4x – 4y + 4 = 0 at
A (1, 1) B (–1, –1) C (1 , 1 ) D None of these
2 2
13. The tangent at (1, 7) to the curve x2 = y – 6 touches the circle x2 + y2 + 16x + 12y + c = 0 at

A (6, 7) B (-6, 7) C (6, –7) D (–6, –7)

14. The slope of the line l1 which is perpendicular to l2 where l2 is the common tangent to the parabolas y = x2
and y = –(x–2)2 is
1 1
A 4 B –4 C D –
4 4
15. The chords of contact of the pairs of tangents drawn from each point on the line 2x + y = 4 to the parabola y2
= –4x pass through the point
1 1 1 1
A (2, –1) B ( , ) C (– ,– ) D (–2, 1)
2 4 2 4
16. If a, b, c are distinct positive real numbers such that the parabolas y2 = 4ax and y2 = 4c (x – b) will have a
common normal, then

b b b b
A 0< <1 B >2 C <0 D 1< <2
a−c a−c a−c a−c
17. Let, e = ecentricity of an ellipse, e = conjugate of e.
0 = golden ratio. If for an ellipse, the distance between the foci is equal to the length of the lotus rectum then
e=
A  B 

C D None of these
2

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x2 y2
18. If the normal at one end of the Latus rectum of an ellipse +
2 b2
= 1 passes through the one end of the
minor axis, then a

A e4 – e2 + 1 = 0 B e2 – e – 1 = 0 C e2 + e + 1 = 0 D e4 + e2 – 1 = 0

19. An ellipse slides bettween two perpendicular straight lines. Then, the locus of its centre is

A a circle B an eclipse C a hyperbola D a circle

1 1
20. If e and e' be the ecentricities of a hyperbola and its conjugate, then +
e2 e′2
A 0 B 1 C 2 D None of these

21. The locus of the centre of a circle which touches given circles externally is,

A a circle B an ellipse C a hyperbola D a pair of straight lines

Assertion Reason based Questions (22-25)

In the following questions, a statement of assertion (A) is followed by a statement of Reason (R). Choice the
correct answer out of the following choices.
(a) both (A) and (R) are true and R is the correct explanation of A
(b) Both (A) and (R) are true but R is not the correct explanation of A
(C) (A) is true but (R) is false.
(D) (A) is false but (R) is true.
22. (A) : The equation of the chord x2 – y2 = 9 which is bisected at (5, –3) is 5x + 3 y – 16 = 0
x2 y2
(R) : The equation of the chord − = 1, bisected at the point (x1, y1) is
a2 b2
xx1 yy1 x 2 y 2
− − 1 = 1 − 1 − 1 ie T = S
a2 b2 a2 b2
A a B b C c D d

23. (A) : The ecentricity of a rectangular hyperbola is 2

b2
(R) : e2 = 1 −
a2
A a B b C c D d

x2 y2
24. (A) : The distance of the point '' on the ellipse +
= 1 from a focus is a(1+ecos)
a 2 b2
(R) : If y = mx + c is a tangent to the ellipse, then c2 = a2m2 + b2
A a B b

C c D d

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25. (A) : The slope of a chord of the parabola y2 = 4ax which is normal at one end and which subtends a right
angle at the origin is 2

(R) : The equation of normal to the parabola y2 = 4ax at (at2, 2at) is y + xt = 2at + at3
A a B b C c D d

Case Study based Questions (26-30)

Consider the curves : c1 : x2 + y2 = 2
c2 : y2 = 8x
The common tangents TPR and TQS are drawn for c1 and c2 curves. The tangents touch the circle at the point
P and Q and parabola at the points R and S.

R
P

B x
T o
x2 + y2 = 2
Q y2 = 8x
S

Based on this answer the following questions : .

2
26. Let y = mx + be the equation of common tangents then m
m
5 1
A ± B 1 C 2 D ±
6 2
27. Coordinate of R is

A (2, 4) B (2, –4)

C (–1, 1) D (–1, –1)

28. Length of PQ is
A 2 unit B 3 unit C 4 unit D 6 unit

29. Distance between PQ and RS is

A 2 unit B 3 unit C 4 unit D 6 unit

30. Area of the trapezium PRSQ is

A 5 sq unit B 10 sq unit C 15 sq unit D 20 sq unit

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SUBJECTIVE
1. Find the number of common tangents to the circles
x2 + y2 – 4x – 6y – 12 = 0 and x2 + y2 + 6x + 18 y + 26 = 0
2. Let c be the circle with centre at (1, 1) and radius 1. If T is the circle centred at (o, y), passing through origin
and touching the circle c externally, then calculate the radius of T.
3. if a circle has two of its diameters alongtthe lines x + y = 5 and x – y = 1 and has area 9 then what will be
equation of the circle?

4. The circle x2 + y2 = 4x + 8y + 5 intersects tthe line 3x – 4y = m at two distinct points. Find he range of m.

5. If the line y = mx + c touches the parabola y2 = 4a (x + a) then what is the value of c interms of a and m?
6. Find the latusrectum of a parabola whose focal chord is PSQ such that SP = 3 and SQ = 2

7. The tangents to the parabola y2 = 4ax at P(t1) and Q (t2) intersect on its axis, Find the value of t1 + t2.

8. Find the angle between the tangents drawn from the point (1, 4) to the parabola y2 = 4 x.
9. Find the number of points with integral coordinates that lie in the interior of the region common to the
circle x2 + y2 = 16 and the parabola y2 = 4x.

10. Find he radius of the largest circle which passes through the focus of the parabola y2 = 4x and contained in
it.

11. If P (x, y), F1 = (3, 0), F2 = (–3, 0) and 16x2 + 25 y2 = 400, Then find the value of PF1 + PF2.

x2 y2
12. Find the slope of a common tangent to the ellipse +
= 1 and concentric circle of radius r.
a 2 b2
13. An ellipse has OB as a semi minor axis, F, F' as its foci and the anglel FBF' is a right angle. Find th
ecentricity of the ellipse.

14. Find the ecentricity of the conic repesented by x2 – y2 – 4 x + 4 y + 16 = 0

15. If a hyperbola passing through the origin has 3x – 4 y – 1 = 0 and 4x - 3y - 6 = 0 as its asymptotes, then find
the equation of its transverse axis.

OBJECTIVE

A N S W E R

1. A 5. C 9. C 13. D 17. A 21. C 25. A 29. B

2. B 6. D 10. D 14. B 18. D 22. D 26. B 30. C
3. C 7. B 11. B 15. D 19. D 23. C 27. A
4. A 8. B 12. C 16. B 20. B 24. B 28. A

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SUBJECTIVE
A N S W E R
LEVEL – 1
1
1. 3 2. 3. x2 + y2 –6x–4y + 4 = 0 4. –35 < m < 15
4

a 24 p
5. c = am + 6. 7. 0 8. 9. – 8
m 5 3

r 2 − b2 1
10. 4 11. 10 12. 13. 14. 2
a2 − r2 2

15. x + y – 5 = 0

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