2D COORDINATE SYSTEM AND LOCUS

1. If a vertex of an equilateral triangle is on origin and second vertex is (4, 0), then its third vertex
is
(a) (2 , ± √ 3) (b) (3 , ± √ 2) (c) (2 , ±2 √ 3) (d) (3 , ± 2 √ 2)
2. The distance of the middle point of the line joining the points (asin θ, 0) and (0,acosθ ) from the
origin is
a 1
a( sinθ +cos θ) a (sin θ+cos θ) a
(a) 2 (b) 2 (c) (d)
3. If the points (1,1), (–1, –1) and (− √3,k ) are vertices of an equilateral triangle then the value of k
will be
(a) 1 (b) –1 (c)√ 3 (d) −√ 3
4. If the points (0, 0), (2, 2 √ 3) and (a, b) be the vertices of an equilateral triangle, then (a, b)=
(a) (0, – 4) (b) (0, 4) (c)(4, 0) (d) (– 4, 0)
5. The point which divides externally the line joining the points (a+b , a−b) and (a−b ,a+b) in the
ratio a :b , is

(a) (a2 −2 ab−b 2 a2+ b2

a−b
,
a−b ) (b) ( a2 −2 ab−b 2 a2−b2
a−b
,
a−b )
( a −2a−bab+b , aa−b )
2 2 2 2
+b
(c) (d) None of these

6. The points which trisect the line segment joining the points (0, 0) and (9, 12) are
(a) (3,4), (6,8) (b) (4,3), (6,8) (c) (4,3), (8,6) (d) (3,4), (8,6)
7. If the point (x, – 1), (3, y), (– 2,3) and (– 3, – 2) be the vertices of a parallelogram, then
(a) x=2 , y=4 (b) x=1 , y=2 (c) x=4 , y=2 (d) None of these
8. The mid-points of sides of a triangle are (2, 1), (–1, –3) and (4,5). Then the coordinates of its
vertices are
(a) (7,9) ,(−3 ,−7) ,(1,1) (b)(−3 ,−7),(1,1),(2,3) (c)
(1,1),( 2,3),(−5,8), (d) None of these

9. Point ( 12 , −134 )divides the line joining the points (3 ,−5) and (−7,2) in the ratio of
(a) 1 : 3 internally (b) 3 : 1internally (c) 1 : 3 externally (d) 3:1 externally
10. The coordinates of the join of trisection of the points (–2, 3), (3, –1) nearer to (–2, 3), is

(b) 3 3 )
(
(a)
(− 13 , 53 ) 4 1
, (− , 2)
(c) 4
3
(d)
( 13 , 53 )
11. If the vertices of a triangle are A(1,4), B(3,0) and C(2,1), then the length of the median passing
through C is
(a) 1 (b) 2 (c) √ 2 (d) √3
12. Three vertices of a parallelogram taken in order are (−1 ,−6), (2 ,−5) and (7,2). The fourth
vertex is
(a) (1, 4) (b) (4, 1) (c)(1, 1) (d) ( 4, 4)
13. P and Q are points on the line joining A (–2, 5) and B (3, 1) such that AP = PQ = QB. Then the
mid-point of PQ is

(a) ( 12 , 3) (b) ( −12 , 4 ) (c) (2,3) (d) (1,4)

14. In what ratio does the y-axis divide the join of (−3 , −4) and (1,−2)
(a) 1 : 3 (b) 2 : 3 (c)3 : 1 (d) None of these
15. (0, –1) and (0, 3) are two opposite vertices of a square. The other two vertices are
(a) (0, 1), (0, –3) (b) (3, –1) (0, 0) (c) (2, 1), (–2, 1) (d) (2, 2), (1, 1)
 2D COORDINATE SYSTEM AND LOCUS

16. The three points (–2,2), (8, –2) and (–4, –3) are the vertices of
(a) An isosceles triangle (b) An equilateral triangle
(c) A right-angled triangle (d) None of these

17. The points ( √a3 , a) ,( 2√3a , 2 a),( √a3 ,3 a) are the vertices of

(a) An equilateral triangle (b)An isosceles triangle

(c) A right-angled triangle (d) None of these

18. The points (a , b),(c , d) and ( kck +la

+l
,
k +l )
kd+lb
are

(a) Vertices of an equilateral triangle (b) Vertices of an isosceles triangle

(c) Vertices of a right-angled triangle (d) Collinear
19. The points (0, 8/3), (1, 3) and (82, 30) are the vertices of
(a) An equilateral triangle (b)An isosceles triangle
(c) A right-angled triangle (d) None of these
20. The points (3a, 0), (0, 3b) and (a, 2b) are
(a) Vertices of an equilateral triangle (b) Vertices of an isosceles triangle
(c) Vertices of a right-angled isosceles triangle (d) Collinear
2
21. The points (−a ,−b) ,(a , b),(a , ab) are
(a) Vertices of an equilateral triangle (b) Vertices of a right-angled triangle
(c) Vertices of an isosceles triangle (d) Collinear
22. The quadrilateral formed by the vertices (–1,1), (0,–3), (5,2) and (4,6) will be
(a) Square (b) Parallelogram (c) Rectangle (d) Rhombus
23. The points A(–4,–1), B (–2,–4), C(4,0) and D(2,3) are the vertices of
(a) Parallelogram (b) Rectangle (c) Rhombus (d)None of these
24. If the vertices of triangle are (0,2), (1,0) and (3,1), then the triangle is
(a) Equilateral (b) Isosceles
(c) Right angled (d) Isosceles right angled
25. The points (−a ,−b),(0,0),(a , b) and (a 2 , ab) are
(a) Collinear (b) Vertices of a rectangle
(c) Vertices of a parallelogram(d) None of these
26. If the vertices of a triangle be (a , 1),(b , 3) and (4 , c) , then the centroid of the triangle will lie
on x-axis, if
(a) a+ c=−4 (b) a+ b=−4
(c) c=−4 (d) b+ c=−4
27. The incentre of the triangle formed by (0, 0), (5,12), (16, 12) is
(a) (7,9) (b) (9,7) (c) (–9, 7) (d) (–7,9)
28. If the coordinates of the points A, B, C, be (4,4), (3,–2) and (3,–16) respectively, then the area of
the triangle ABC is
(a) 27 (b) 15 (c)18 (d) 7
29. The area of the triangle formed by the points (a , b+ c),(b , c +a) ,(c , a+b) is
(a) abc (b) a 2+ b2+c 2
(c) ab+ bc+ ca (d) 0
30. The vertices of the triangle ABC are (2,1), (4,3) and (2,5). D, E , F are the mid-points of the sides.
The area of the triangle DEF is
(a) 1 (b) 1.5 (c)3 (d) 4
31. If points (5, 5), (10, k) and (–5, 1) are collinear, then k =
(a) 3 (b) 5 (c)7 (d) 9
 2D COORDINATE SYSTEM AND LOCUS

32. The orthocenter of the triangle with vertices (–2, –6), (–2, 4) and (1, 3) is
(a) (–3, 1) (b) (–1,1/3)
(c) (1, 3) (d) None of these
33. Orthocenter of the triangle whose vertices are (0, 0) (3, 0) and (0, 4) is
(a) (0, 0) (b) (1, 1) (c) (2, 2) (d) (3, 3)

LOCUS
1. The locus of the moving point P, such that 2PA = 3PB where A is (0,0) and B is (4,–3), is
(a) 5 x 2−5 y 2−72 x +54 y +225=0(b)5 x 2−5 y 2+72 x +54 y +225=0
(c) 5 x 2+5 y 2 +72 x+54 y+225=0 (d)5 x 2+5 y 2−72 x +54 y +225=0
2. A point moves such that the sum of its distances from two fixed points (ae,0) and (–ae,0) is
always 2a. Then equation of its locus is
2 2 2 2
x y x y
(a) 2
+ 2 2
=1 (b) 2 − 2 2
=1
a a (1−e ) a a (1−e )
2 2
x y
(c) 2 2
+ 2
=1 (d) None of these
a (1−e ) a
3. A point moves in such a way that its distance from (1,–2) is always the twice from (–3,5), the
locus of the point is
(a) 3 x 2+ y2 +26 x +44 y−131=0
(b) x 2+ 3 y2 −26 x+ 44 y−131=0
(c) 3( x 2 + y 2)+26 x−44 y +131=0
(d) None of these
4. A point moves in such a way that its distance from origin is always 4. Then the locus of the
point is
(a) x 2+ y 2=4 (b) x 2+ y 2=16
(c) x 2+ y 2=2 (d) None of these
5. If A(−a , 0) and B(a , 0) are two fixed points, then the locus of the point on which the line AB
subtends the right angle, is
(a) x 2+ y 2=2 a2 (b) x 2− y 2=a2
(c) x 2+ y 2+ a2=0 (d) x 2+ y 2=a2
6. The locus of P such that area of Δ PAB=12sq. units, where A(2,3) and B(−4,5) is
(a) (x +3 y−1)(x+3 y −23)=0
(b) (x +3 y+ 1)(x +3 y−23)=0
(c) (3 x+ y−1)(3 x+ y −23)=0
(d) (3 x+ y+ 1)(3 x + y +23)=0
7. If A( cos α , sin α ) , B (sin α ,−cos α ) , C(1 , 2) are the vertices of a Δ ABC , then as α varies, the
locus of its centroid is
(a) x 2+ y 2−2 x −4 y +1=0 (b)3( x 2 + y 2)−2 x−4 y +1=0
(c) x 2+ y 2−2 x −4 y +3=0 (d) None of these
8. The locus of a point whose difference of distance from points (3, 0) and (–3,0) is 4, is
x2 y2
− =1 x2 y 2 x2 y2
− =1
x2 y2
− =1
− =1
(a) 4 5 (b) 5 4 (c) 2 3 (d) 3 2
9. If the distance of any point P from the point A(a+b , a−b) and B(a−b , a+ b) are equal, then
the locus of P is
(a) x− y =0 (b) ax +by =0
(c) bx−ay=0 (d) x + y=0
 2D COORDINATE SYSTEM AND LOCUS

10. What is the equation of the locus of a point which moves such that 4 times its distance from the
x-axis is the square of its distance from the origin
(a) x 2+ y 2−4 y=0 (b) x 2+ y 2−4∨ y ∨¿ 0
2 2
(c) x + y −4 x =0 (d) x 2+ y 2−4∨x∨¿ 0
11. Let A(2 ,−3) and B(−2,1) be vertices of a triangle ABC. If the centroid of this triangle moves
on the line 2 x+3 y =1, then the locus of the vertex C is the line
(a) 3 x−2 y=3 (b) 2 x−3 y=7
(c) 3 x+ 2 y =5 (d) 2 x+3 y =9