Point_1

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POINT LEVEL-1

1. If A (0, 0) and B (12, 0) are two points, then the co-ordinate of a point C on line joining AB dividing it in the ratio 5 : 1

(externally) are : (A) (10, 0) (B) (15, 0) (C) (18, 0) (D) none of these. 2. If orthocenter and circumcentre of triangle are respectively (1, 1) and (3, 2), then the co-ordinates of its centroid are :

(A)

3

5,

3

7 (B)

3

7,

3

5 (C) (7, 5) (D) none of these.

3. The extremities of a diagonal of a parallelogram are the points (1, - 2) and (- 4, 3). If third vertex is (- 1, 0). Then the co-ordinates of fourth vertex are :

(A) (0, 1) (B) (- 2, - 1) (C) (- 2, 1) (D) none of these. 4. The triangle with vertices A (2, 7), B (4, y) and C (- 2, 6) is right angled if :

(A) 1−=y (B) 0=y (C) 1=y (D) none of these.

5. Circumcentre of triangle whose vertices are (0, 0), (3, 0) and (0, 4) is :

(A)

2 ,

2

3 (B)

2

3 ,2 (C) (0, 0) (D) none of these.

6. If area of triangle with vertices (0, 0), (0, 6) and (α, β) is 15 sq. units, then :

(A) α = ± 5, β = 5 (B) α = ± 10, β = 5 (C) α = ± 5, β = 2 (D) α = ± 5, β can take any real value

7. Vertices of a ∆ABC are A (2, 2), B (- 4, - 4), C (5, - 8). Then length of the median through C is :

(A) 65 (B) 117 (C) 85 (D) 113

8. The orthocenter of the triangle formed by (2, 0), (2, 5) and (5, 0) is : (A) (2, 5) (B) (5, 2) (C) (5, 0) (D) (2, 0) 9. The co-ordinates of circumcentre of the triangle with vertices (8, 6), (8, -2) and (2, - 2) are :

(A)

3

2 ,6 (B) (8, 2) (C) (5, - 2) (D) (5, 2)

10. If two vertices of an equilateral triangle are (0, 0) and (3, 3 3 ), then the third vertex is :

(A) (3, - 3) (B) (- 3, 3) (C) (- 3, 3 3 ) (D) none of these.

11. If P (1, 2), Q (2, 3), R (5, 7) and S (a, b) are the vertices of a parallelogram PQRS, then :

(A) 6,4 == ba (B) 6,2 == ba (C) 8,4 == ba (D) 3,4 == ba

12. If origin is shifted to (7, - 4), then point (4, 5) shifted to : (A) (- 3, 9) (B) (3, 9) (C) (11, 1) (D) none of these. 13. The points (1, - 2), (3, 0), (1, 2) and (- 1, 0) are the vertices of a : (A) parallelogram (B) rectangle (C) square (D) cyclic quadrilateral 14. If the area of a triangle with vertices (4, 0), (1, 1) and (3, a) be 2, then ‘a’ is equal to :

(A) – 1 or 3

5 (B)

3

5 or 1 (C) 2 or – 1 (D) 3 or

3

5

15. The co-ordinates of the middle points of the sides of a triangle are (4, 2), (3, 3) and (2, 2), then co-ordinates of centroid are:

(A) (3, 7/3) (B) (3, 3) (C) (4, 3) (D) none of these 16. The incentre of the triangle whose vertices are (– 36, 7), (20, 7) and (0, – 8) is:

(A) (0, – 1) (B) (– 1, 0) (C)

1 ,

2

1 (D) none of these

17. The medians of a triangle meet at (0, – 3) and two vertices are at (– 1, 4) and (5, 2). Then the third vertex is at: (A) (4, 15) (B) (– 4, – 15) (C) (– 4, 15) (D) (4, – 15) 18. If the orthocenter and centroid of a triangle are (– 3, 5) and (3, 3), then its circumcentre is: (A) (6, 2) (B) (3, – 1) (C) (– 3, 1) (D) (– 3, 5) 19. Let P and Q be the points on the line joining A (– 2, 5) and B (3, 1) such that A P = P Q = Q B. Then the mid-point of P Q

is:


(A)

3 ,

2

1 (B)

− 4 ,

2

1 (C) (2, 3) (D) (– 1, 4)

20. A triangle ABC right angled at A has points A and B as (2, 3) and (0, – 1) respectively. If BC = 5 units, then the point C

is: (A) (– 4, 2) (B) (4, 2) (C) (3, – 3) (D) (0, – 4)

21. The locus of a point P which divides the line joining (1, 0) and (2 cos θ, 2 sin θ) internally in the ratio 2 : 3 for all θ is a: (A) straight line (B) circle (C) pair of straight lines (D) parabola 22. The vertices of a triangle are A (0, 0), B (0, 2) and C (2, 0). The distance between circumcentre and orthocentre is:

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

1 (D) none of these

23. The incentre of the triangle with vertices ),3,1( (0, 0) and (2, 0) is:

(A)

2

3,1 (B)

3

1,

3

2 (C)

2

3,

3

2 (D)

3

1,1

24. The medians AD and BE of a triangle with vertices A (0, b), B (0, 0) and C (a, 0) are perpendicular to each other if:

(A) ab = 1 (B) a = ± b2 (C) 2

ba = (D)

2

ab =

25. If the point )](),([(121121

yytyxxtx −+−+ divides the join of ),( and ),(2211

yxyx internally then:

(A) t < 0 (B) 0 < t < 1 (C) t > 1 (D) t = 1 26. If P (1, 2), Q (4, 6), R (5, 7) and S (a, b) are the vertices of a parallelogram PQRS, then: (A) a = 2, b = 4 (B) a = 3, b = 4 (C) a = 2, b = 3 (D) a = 3, b = 5

27. 0

1

1

1

33

22

11

=

yx

yx

yx

is the condition that the points iyx ii ),,( = 1, 2, 3:

(A) from an equilateral triangle (B) are collinear

(C) from a right angled triangle (D) ),(22

yx is the mid-point of the line joining ),(),,(3311

yxyx I

28. The point P is equidistant from A (1, 3), B (– 3, 5), C (5, – 1) then PA is:

(A) 5 (B) 55 (C) 25 (D) 105

29. The X – co-ordinate of the in centre of the triangle where the mid-points of the sides are (0, 1), (1, 1) and (1, 0) is:

(A) 2 + 2 (B) 1 + 2 (C) 2 – 2 (D) 1 – 2

30. The points (x + 1, 2), (1, x + 2),

++ 1

2,

1

1

xx are collinear then x is equal to:

(A) 4 (B) 0 (C) – 2 (D) none of these 31. If d [(x, – 1), (3, 2)] = 5, then x is: (A) 2 (B) 7 (C) – 2 (D) 0

32. The polar coordinates of the vertices of a triangle are (0, 0), (3, π/2) and (3, π/6). Then the triangle is (A) right angled (B) isosceles (C) equilateral (D) none of these

33. The point ),(),,(),,( bacacbcba +++ are

(A) vertices of an equilateral triangle (B) collinear (C) concyclic (D) none of these 34. The area of the pentagon whose vertices are (4, 1), (3, 6), (- 5, 1), (- 3, - 3) and (- 3, 0) is (A) 30 unit

2 (B) 60 unit

2 (C) 120 unit

2 (D) none of these

35. The coordinates of the point on the x-axis which is equidistant from the points (- 3, 4) and (2, 5) are

(A) (20, 0) (B) (- 23, 0) (C)

0,

5

4 (D) none of these

36. If P (1, 2), Q (4, 6), R (5, 7) and S (a, b) are the vertices of a parallelogram ,PQRS then

(A) 4,2 == ba (B) 4,3 == ba (C) 3,2 == ba (D) 5,3 == ba

37. QP, and R are three collinear points. The coordinates of P and R are (3, 4) and (11, 10) respectively, and PQ is

equal to 2.5 units. Coordinates of Q are


(A)

2

11,5 (B)

2

5,11 (C)

−

2

11,5 (D)

−

2

11,5

38. Vertices of a quadrilateral ABCD are A (0, 0), B (3, 4), C (7, 7) and D (4, 3). Then quadrilateral ABCD is

(A) Square (B) Parallelogram (C) Rectangle (D) Rhombus

39. If A and B are the points (- 3, 4) and (2, 1). Then the coordinates of the point C on AB produced such that

BCAC 2= are

(A) (- 1/2, 5/2) (B) (7, - 2) (C) (3, 7) (D) (2, 4)

40. If G is the centroid and I is the incentre of the triangle with vertices A (- 36, 7), B (20, 7) and C (0, - 8), then value of

GI is

(A) 205 (B) 2052

1 (C) 205

3

1 (D) 205

3

2

Answers Level - 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

b a c a a d c d d c a a c a a b b a a b

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

b b d b b c d c b b c b a d c a d b c


LEVEL-2

1. If (α, β), ),( yx and (p, q) are the coordinates of the circumcentre, the centorid and the orthocenter of a triangle, then

(A) 3

2,

3

2 qy

px

+=

+=

βα (B)

3

2,

3

2 qy

px

+=

−=

βα

(C) 3

2,

3

2 qy

px

+=

+=

βα (D)

3

2,

3

2 qy

px

−=

−=

βα

2. If area of the pentagon ABCDE be 2

45 where A = (1, 3), B = (- 2, 5), C = (- 3, - 1), D = (0, - 2) and E = (2, t), then

value of t is

(A) 1 (B) – 1 (C) 2 (D) – 2 3. If A and B are two points having co-ordinates (3, 4) and (5, - 2) respectively and P is a point such that PA = PB and

area of triangle PAB = 10 sq. units then the co-ordinates of P are : (A) (7, 4) and (13, 2) (B) (7, 2) and (1, 0) (C) (2, 7) and (4, 13) (D) none of these.

4. In the ∆ABC, the coordinates of B are (0, 0), AB = 2, ∠ABC = 3

π and the middle point of BC has the coordinates (2, 0).

The centroid of the triangle is

(A)

2

3,

2

1 (B)

3

1,

3

5 (C)

+

3

1,

3

34 (D) none of these

5. Let P = (1, 1) and Q = (3, 2). The point R on the x-axis such that PR + RQ is the minimum is

(A)

0,

3

5 (B)

0,

3

1 (C) (3, 0) (D) none of these

6. If the origin is shifted to (2, – 1), then the transformed equation of 012422 =++−+ yxyx is:

(A) 2 2

4 0x y+ − = (B) 2 2

0x y+ = (C) 822 =+ yx (D) none of these

7. A and B are two points whose coordinates are )2,(2 amam and

−

m

a

m

a 2,

2 and S is the point (a, 0). Value of

SBSA

11+ is

(A) a (B) a

1 (C) a2 (D)

a

2

8. The acute angle θ through which the co-ordinate axes should be rotated for the point A(2, 4) to attain the new abscissa 4 is given by

(A) 4/3tan =θ (B) 8/7tan =θ (C) 6/5tan =θ (D) 5/4tan =θ

9. P (3, 1), Q (6, 5) and R (α, β) are three points such that area of ∆ 7=RPQ and ∠PRQ is a right angle, then the

number of such point R is (A) 0 (B) 1 (C) 2 (D) 4

10. The points (α, β), (γ, δ), (α, δ) and (γ, β), where α, β, γ, δ are different real numbers are (A) vertices of a square (B) vertices of a rhombus (C) collinear (D) concyclic

11. If the straight line joining the point (a, b) and (c, d) subtends an angle θ at the origin. Value of cos θ is

(A)

)()(2222 dcba

bcab

++

+ (B)

)()(2222

dcba

bcab

++

−

(C)

)()(2222 dcba

bdac

++

+ (D)

)()(2222

dcba

bdac

++

−

12. The number of integer values of ,m for which the x-coordinate of the point of intersection of the lines 943 =+ yx and

1+= mxy is also in integer is :

(A) 2 (B) 0 (C) 4 (D) 1


13. The line joining the points ),(11

yx and ),(22

yx subtends a right angle at the point (1, 1) if

(A) 21212121

yyxxyyxx +++=+ (B) 12121212

−+++=+ yyxxyyxx

(C) 221212121

−+++=+ yyxxyyxx (D) 221212121

−+++=+ yyxxyyxx

14. ABC is an equilateral triangle such that the vertices B and C lie on two parallel lines at a distance 6. If A lies between the parallel lines at a distance 4 from one of them then the length of a side of the equilateral triangle is

(A) 8 (B) 3

88 (C)

3


15. Base of an isosceles triangle is of length a2 and the length of the altitude dropped to the base is .p The distance from

the mid-point of the base to the side of the triangle is

(A)

)(22 pa

a

+ (B)

22 ap

p

+ (C)

22 pa

ap

+ (D) none of these

16. Without changing the direction of coordinate axes, origin is transferred to (h, k) so that the linear (one degree) terms are

eliminated from the equation .076422 =++−+ yxyx Then the point (h, k) is:

(A) (3, 2) (B) (– 3, 2) (C) (2, – 3) (D) (2, 3) 17. If two vertices of an equilateral triangle have integral coordinates then the third vertex will have (A) integral coordinates (B) coordinates which are rational (C) at least one coordinate irrational (D) coordinates which are irrational

18. If points A ),(),,(2211

yxByx and ),(33

yxC are such that 321

,, xxx and 321

,, yyy are in A.P., then :

(A) A, B and C re concyclic points (B) A, B and C are collinear points (C) A, B and C are vertices of an equilateral triangle (D) none of these.

19. If points ),,(11

yxA ),(22

yxB and ),(33

yxC are such that 321

,, xxx and 321

,, yyy are in G.P. with same common

ratio, then : (A) A, B and C are concyclic points (B) A, B and C are collinear points (C) A, B and C are vertices of an equilateral triangle (D) none of the above 20. If (a, 0) and (0, b) are two opposite vertices of a diagonal of a square, then one of the remaining two vertices is :

(A) (0, 0) (B) (a, b) (C)

2,

2

ba (D)

++

2,

2

baba

21. If a point P (4, 3) is rotated through an angle 45o in anticlockwise direction about origin, then co-ordinates of P is new

position are :

(A)

2

7,

2

1 (B)

−−

2

1,

2

7 (C)

−

2

7,

2

1 (C)

−

2

7,

2

1

22. The point (4, 1) undergoes the following three transformations successively (i) reflection about the line .xy =

(ii) translation through a distance 2 units along the positive direction of x-axis.

(iii) rotation through an angle 4

π about the origin in the anticlockwise direction.

(iv) The final position of the point is given by the co-ordinates.

(A)

2

7,

2

1 (B) (- 2, )27 (C)

−

2

7,

2

1 (D) (2, )27

23. P, Q, R and S are the points on line joining the points ),( xaP and ),( ybT such that ,STRSQRPQ === then

++

8

35,

8

35 yxba is the mid-point of :

(A) PQ (B) QR (C) RS (D) ST 24. The points with the co-ordinates (2a, 3a), (3b, 2b) and (c, c) are collinear: (A) for no value of a, b, c (B) for all values of a, b, c

(C) iff a, bc

,5

are in H.P. (D) iff a, bc,5

2 are in H.P.

25. Locus of the centroid of the triangle whose vertices are (a cos t, a sin t), (b sin t, - b cos t) and (1, 0) where t is a parameter is:

(A) 2222

)3()13( bayx −=++ (B) 2222

)3()13( bayx −=+−


(C) 2222

)3()13( bayx +=+− (D) 2222

)3()13( bayx +=++

26. If the equation of the locus of a point equidistant from the points ),( and ),(2211

baba is

0)()(2121

=+−+− cybbxaa then the value of c is:

(A) 2

2

2

2

2

1

2

1baba −−+ (B) )(

2

1 2

1

2

1

2

2

2

2 baba −−+

(C) 2

2

2

1

2

2

2

1bbaa −+− (D) )(

2

1 2

2

2

1

2

2

2

1 bbaa +++

27. If a point (x, y) = (tan θ + sin θ, tan θ – sin θ), then the locus of (x, y) is:

(A) ( ) ( ) 13/223/22 =+ xyyx (B) xyyx 4

22 =−

(C) xyyx 1222 =− (D) xyyx 16)(

222 =−

28. If

−==

t

a

t

aBatatA

2,),2,(

2

2 and )0,(aS = , then aSA 2, and SB are in

(A) A.P. (B) G.P. (C) H.P. (D) none of these

29. The coordinates of P, Q, R and S are (6, 3), (- 3, 5), (4, - 2) and (x, 3x), respectively. If the areas of ∆ PQR is twice that

of ∆ ,SQR then one of the values of x will be

(A) 11/8 (B) – 11/8 (C) 3 (D) – 3

30. The points (1, 3) and (5, 1) are two opposite vertices of a rectangle. The other two vertices lie on the line .2 λ+= xy

Then value of λ is (A) – 4 (B) 4 (C) 3 (D) – 2

31. ABC is a variable triangle with the fixed vertex C (1, 2) and BA, having the coordinates (cos θ, sin θ), (sin θ, - cos θ)

respectively, where θ is a parameter. The locus of the centroid of ∆ ABC is

(A) 0222 =+−−+ yxyx (B) 0142)(3

22 =+−−+ yxyx

(C) 0142)(322 =−−++ yxyx (D) none of these

32. If P (1, 0), Q (- 1, 0) and R (2, 0) are three given points, then the locus of point S satisfying the relation 222

2SPSRSQ =+ , is :

(A) a straight line parallel to x-axis (B) a circle through the origin (C) circle with centre at the origin (D) a straight line parallel to y-axis

33. The three distinct points )12 ,(),2,(),1 ,(22 +++ kkkkkkk are collinear for :

(A) All real values of k (B) no value of k (C) Exactly two values of k (D) None of these 34. If (2, 0) and (3, 1) are the vertices of base of a triangle then the locus of its vertex if area of triangle is unity is : (A) x – y = 0 (B) x – y – 4 = 0 (C) (x – y) (x – y – 4) = 0 (D) none of these

35. If the distance of any point (x, y) from the origin is defined as d (x, y) = max, { | x |, | y |}. Then the locus of ayxd =),(

non-zero constant, is : (A) A circle (B) A straight line (C) Square (D) Triangle 36. The locus of a point which moves such that the sum of the squares of its distances from the three vertices of a triangle is

constant is a circle whose centre is at the : (A) Centroid of the triangle (B) Orthocentre (C) Incentre (D) None of these

37. The line joining (5, 0) to (10 cos θ, 10 sin θ) is divided internally in the ratio 2 : 3 at P. If θ vertices, then the locus of P is : (A) a pair of straight lines (B) a circle (C) a straight line (D) none of these 38. A straight rod of length 9 units slides with its ends A, B always on the X and Y-axis respectively. Then the locus of

centroid of ∆OAB is :

(A) 322 =+ yx (B) 9

22 =+ yx (C) 122 =+ yx (D) 81

22 =+ yx

39. Given the points A(0, 4) and B(0, - 4), the equation of the locus of the point P(x, y) such that |AP – BP| = 6 is

(A) 279

22

=−yx

(B) 179

22

=−yx

(C) 159

22

=−yx

(D) None of these

40. The locus of the point of intersection of lines θθ cos ,sin24 ayxayx =−=+ where θ is a variable parameter is :


(A) 222

205 ayx =+ (B) 222

2205 ayx =+ (C) 222

3205 ayx =+ (D) 222

4205 ayx =+

Level - 3 More than one option may be correct

1. The points )3,12(),1,1( ++ pp and )2,22( pp + are collinear if

(A) 1−=p (B) 2/1=p (C) 2=p (D) 2/1−=p

2. The points A(- 3, 4), B(4, 6) and C(- 1, - 3) are the vertices of (A) a right-angled triangle (B) an isosceles triangle (C) an acute-angled triangle (D) an obtuse-angled triangle. 3. The coordinates of A, B, C and D are (6, 3), (- 3, 5), (4, - 2) and (x, 3x), respectively. If the area of triangle ABC is twice

that of triangle DBC, the value of x can be (A) – 3/8 (B) – 3 (C) 11/8 (D) 4. 4. If the coordinates of the vertices of a triangle are rational numbers then which of the following points of the triangle will

always have rational coordinates? (A) Centroid (B) Incentre (C) Circumcentre (D) Orthocentre 5. Two consecutive vertices of a rectangle of area 10 unit

2 are (1, 3) and (- 2, - 1). Other two vertices are

(A)

−

−

5

1,

5

18,

5

21,

5

3 (B)

−−

−

5

11,

5

2,

5

21,

5

3

(C)

−−

5

9,

5

13,

5

11,

5

2 (D)

−

5

1,

5

18,

5

9,

5

13

6. The ends of a diagonal of a square are (2, - 3) and (- 1, 1). Another vertex of the square can be

(A)

−−

2

5,

2

3 (B)

2

1,

2

5 (C)

2

5,

2

1 (D) none of these

7. If each of the vertices of a triangle has integral coordinates then the triangle may be (A) right angled (B) equilateral (C) isosceles (D) none of these 8. If (- 1, 2), (2, - 1) and (3, 10) are any three vertices of a parallelogram then the fourth vertex (a, b) will be such that

(A) 0,2 == ba (B) 0,2 =−= ba (C) 6,2 =−= ba (D) 2,6 −== ba

9. The centroid and a vertex of an equilateral triangle are (1, 1) and (1, 2) respectively. Another vertex of the triangle can be

(A)

−

2

1,

2

32 (B)

+

2

1,

2

332 (C)

+

2

1,

2


10. The point P divides the join A (- 5, 1) and B (3, 5) in the ratio λ : 1 internally. Points Q and R are (1, 5) and (7, 2)

respectively. Area of ∆ PQR is 2, for λ =

(A) 19/5 (B) 31/9 (C) 23 (D) 19

11. The coordinates of two points A and B are (3, 4) and (5, -2) respectively and a point P is such that BPAP = and

∆ .10=PAB Then P is

(A) (1, 0) (B) (2, 7) (C) (4, 13) (D) (7, 2)

12. ABC is an isosceles triangle. If the coordinates of the base are B(1, 3) and C(- 3, 7), the coordinates of vertex A can be

(A) (6, 5/6) (B) (5/6, 6)

(C) (- 7, 1/8) (D) any point on the line 04386 =+− yx

13. A point ),( yxP moves in such a way that the area of the triangle formed with A(1, - 1) and B(5, 2) is of magnitudes 5

units, then locus of P is

(A) 01743 =−− yx (B) 0343 =+− yx (C) 01734 =−+ yx (D) 0334 =++ yx

14. In a triangle ABC, AB = AC and the co-ordinates of B and C are (1, 3) and (-2, 7) respectively.. The co-ordinates of A can be

(A)

− 5,

2

1 (B)

−

8

1,7 (C) (1, 6) (D)

6,

6

5

15. The points )0,4(),4,2(),1,4( CBA −−−− and )3,2(D are the vertices of a

(A) parallelogram (B) rectangle (C) rhombus (D) square.

16. If the point ),( yxP is equidistant from the points ),( abbaA −+ and ),,( babaB +− then


(A) byax = (B) aybx = (C) )(222 byaxyx +=− (D) P can be (a, b).

17. The coordinates of three points O, A and B are (0, 0), (0, 4) and (6, 0), respectively. If a point P moves so that the

area of ∆POA is always twice the area of ∆POB, then P lies on

(A) 03 =− yx (B) 03 =+ yx (C) 043 =+ yx (D) .043 =− yx

18. If the points (2a, a), (a, 2a) and (a, a) enclose a triangle of area 18 sq. units, the centroid of the triangle is (A) (- 8, - 8) (B) (- 4, - 4) (C) (4, 4) (D) (8, 8) 19. ABC is an isosceles triangle whose base is BC. If B and C are (a + b, b – a) and (a – b, a + b), then co-ordinates of A may

be

(A) (a, b) (B) (b, a) (C)

a

b

b

a, (D)

a

b,1

20. If two vertices of an equilateral triangle are (1, 1) and (-1, -1), the co-ordinates of the third vertex may be

(A) ( )3,3 −− (B) ( )3,3− (C) ( )3,3 − (D) ( )3,3

Level -4

1. Prove that the area of the triangle formed by joining the mid-point of the sides of ABC∆ is a quarter of the area of

∆ ABC.

2. If D is the mid-point of BC in ∆ ABC, prove that AB2 + AC

2 = 2(AD

2 +BD

2).

3. Prove using methods of co-ordinate geometry, that if the mid-point of the sides of a quadrilateral are joined in order, we get a parallelogram.

4. Prove that not both co-ordinate of all the vertices of an equilateral triangle can be rational numbers.

5. Origin is the circumcentre of the triangle with vertices A(x1,x1tanθ 1), B(x2, x2tanθ 2),

C(x3,x3tanθ 3), (0<θ i<2

π , xi > 0, i = 1,2,3). If the centroid of ∆ABC is (x,y) prove that

321

321

sinsinsin

coscoscos

θθθ

θθθ

++

++=

y

x.

6. Find the circumcentre and the circumradius of the triangle with vertices (3, 0), (-1, -6) and (4, -1).

7. A is (-4, 6), B is (2, -2) and C∈{(x, y) 012 =+− yx }. Prove that the equation of the locus of the centroid of ∆ ABC is

2x – y +3 =0.

8. A is (2 2 , 0) and B is (-2 2 , 0). If PBAP − = 4, find the equation of the locus of P.

9. If G is the centroid of ∆ ABC, prove : AB2 +BC

2 + AC

2 = 3(GA

2 + GB

2 +GC

2)

10. Show that the centroid of any given triangle is the same as the centroid of the triangle whose vertices are the mid-points of the sides of the given triangle.

11. Show that for the triangle with vertices (1, a), (2, b), (c2

, -3) the centroid can never be on the Y-axis.

12. Show that the area of the triangle whose vertices are (t,t – 2),(t + 2, t +2),(t + 3, t) does not depend on t.

13. If the origin is shifted to (2, -1), obtain the transformed form of the equation , x2

+ y2

- 4x + 2y + 1 =0.

14. Points A and B have co-ordinates (2, 2) and (6, 6). Find the co-ordinates of a point P such that PA = PB and the area of

∆ PAB = 4. 15. A(1, 1) and B(2, - 3) are two fixed points. Find the equation of the locus of P such that PA = 3PB. 16. If G is the centroid of a triangle ABC and O any other point, prove that

(i) 222222

)(3 ABCABCGCGBGA ++=++

(ii) .32222222 GOGCGBGAOCOBOA +++=++

17. If the point (x, y) is equidistant from the point (a + b, b – a) and (a – b, a + b), prove that .aybx =

18. Find the distance between the points

3,12

π and

−

6,9

π.

19. Find the co-ordinates of the point which divides externally the joining the points (6, - 9) and (4, 6) in the ratio 3 : 4.

20. In what ratio does the (a) point (- 5, 3) divide the join of (- 3, - 1) and (- 8, 8)?


(b) point (1, 12) divide the join of (5, 6) and (7, 3)? (c) point (- 5, - 20) divide the join of (4, 7) and (1, - 2)? (d) x-axis divide the join of (2, - 4) and (- 3, 6)? Find the point of section also. (e) y-axis divide the join of (7, 3) and (- 5, - 12)?

21. Prove that the straight line 02 =+− xy cuts the straight line joining (3, - 1) and (8, 9) in the ratio 2 : 3.

22. Transform to parallel axes through the point (1, - 2) the equation 08442 =++− yxy

23. Find the coordinates of the point on transferring the origin to which the equation 05644322 =+−−++ yxyxyx

does not contain linear terms in x and y. Also find the new equation. 24. Show that if the origin be transferred to (0, 1) and the axes rotated through 45

o, the equation

=−−++− 710252522 yxyxyx 0 referred to new axes becomes 1

23

22

=+yx

25. The equation 050221832322 =+−−++ yxyxyx is transformed to 124

22 =+ yx when referred to rectangular

axes through the point (2, 3). Find the inclination of the letter axes to the former.

Answers Level - 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

b a c a a d c d d c a a c a a b b a a b

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

b b d b b c d c b b c b a d c a d b c

Level – 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

a b b b a a b a a d c a c c c c c b b d

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

a c b d c b d c a a b d a c c a b b b d Level – 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

cd ab ac acd ac ab acd bd ac ac ad bcd ad abd ab

16 17 18 19 20

b ab ad ad bc Level – 4

15. 011556348822 =++−+ yxyx 18. 15 19. (12, - 54)

20.(a) 2 : 3 Internally (b) 2 : 3 Externally (c) 3 : 2 Externally (d) 2 : 3 (e) 7 : 3,

−

4

23,0 22. xy 4

2 =

23. (2, 0), .014322 =+++ yxyx 25. 45

o

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