CBSE Class 10 Coordinate geometry Sure Shot Questions Set 07

Question. The points \( (k – 1, k + 2), (k, k + 1), (k + 1, k) \) are collinear for
(a) any value of k
(b) \( k = –1/2 \) only
(c) no value of k
(d) integral values of k only
Answer: (a) any value of k

Question. The area of the triangle formed by the points \( (k, 2 – 2k), (– k + 1, 2k) \) and \( (– 4 – k, 6 – 2 k) \) is 70 units. For
(a) four real values of k
(b) no integral value of k
(c) two integral values of k
(d) only one integral value of k
Answer: (d) only one integral value of k

Question. The quadrilateral ABCD formed by the points A (0, 0), B (3, 4), C (7, 7) and D (4, 3) is a
(a) rectangle
(b) square
(c) rhombus
(d) parallelogram
Answer: (c) rhombus

Question. The triangle with vertices A (2, 7), B (4, y) and C(– 2, 6) is right angled at A if
(a) \( y = –1 \)
(b) \( y = 0 \)
(c) \( y = 1 \)
(d) None
Answer: (a) \( y = –1 \)

Question. The join of the points \( (– 3, – 4) \) and \( (1, – 2) \) is divided by y-axis in the ratio.
(a) 1 : 3
(b) 2 : 3
(c) 3 : 1
(d) 3 : 2
Answer: (c) 3 : 1

Question. The following points A(–1, 0), B(3,1), C(2,2) and D(–2,1) taken in order form a
(a) rectangle
(b) parallelogram
(c) square
(d) None
Answer: (b) parallelogram

Question. Find the value of k, if the distance between the points (1,4) and (k, 1) is 5.
(a) 3 or 5
(b) – 3 or 5
(c) 3 or – 5
(d) None
Answer: (b) – 3 or 5

Question. The centroid of a triangle lies at the origin and the coordinates of its two vertices are \( (–8, 7) \) and \( (9, 4) \). The area of the triangle is
(a) 95/6
(b) 285/2
(c) 190/3
(d) 285
Answer: (b) 285/2

Question. The mid points of the sides AB and AC of a triangle ABC are \( (2, – 1) \) and \( (– 4, 7) \) respectively, then the length of BC is
(a) 10
(b) 20
(c) 25
(d) 30
Answer: (b) 20

Question. If the vertices of a triangle ABC are A \( (– 4, – 1) \), B (1, 2) and C (4, – 3), then the coordinates of the circumcentre of the triangle are
(a) \( (1/3, – 2/3) \)
(b) \( (0, – 4) \)
(c) \( (0, – 2) \)
(d) \( (– 3/2, 1/2) \)
Answer: (c) \( (0, – 2) \)

Question. The extremities of a diagonal of a parallelogram are the points \( (3, – 4) \) and \( (– 6, 5) \). If third vertex is \( (– 2, 1) \) then the coordinates of the fourth vertex are
(a) \( (1, 0) \)
(b) \( (0, 0) \)
(c) \( (1, 1) \)
(d) None
Answer: (d) None

Question. The number of lines that can be drawn through the point \( (4, – 5) \) at a distance 12 from the point \( (– 2, 3) \) is
(a) 0
(b) 1
(c) 2
(d) Infinite
Answer: (a) 0

Question. If O is the origin and the coordinates of A and B are \( (x_1, y_1) \) and \( (x_2, y_2) \) respectively then \( OA \times OB \cos \angle AOB \) is equal to
(a) \( x_1 y_1 + x_2 y_2 \)
(b) \( x_1 x_2 + y_1 y_2 \)
(c) \( x_1 y_2 + x_2 y_1 \)
(d) \( x_1 x_2 – y_1 y_2 \)
Answer: (b) \( x_1 x_2 + y_1 y_2 \)

Question. Write an equation for a line passing through the points (c, 2b) and (c,3b).
(a) \( y = cx – b \)
(b) \( y = – cx + b \)
(c) \( x = 2b \)
(d) \( x = c \)
Answer: (d) \( x = c \)

Question. The line \( \frac{x}{3} + \frac{y}{4} = 1 \) meets the axis of y and axis of x at A and B respectively. A square ABCD is constructed on the line segment AB away from the origin, the coordinates of the vertex of the square farthest from the origin are
(a) (7, 3)
(b) (4, 7)
(c) (6, 4)
(d) (3, 8)
Answer: (b) (4, 7)

Question. ABCD is a quadrilateral. P (3, 7) and Q (7, 3) are the middle points of the diagonals AC and BD respectively. The coordinates of the mean point (or the centre of mean position) of the vertices of the quadrilateral are
(a) (0, 0)
(b) (3, 3)
(c) (5, 5)
(d) (7, 7)
Answer: (c) (5, 5)

Question. The area of a triangle, two of whose vertices are (2, 1) and \( (3, – 2) \) is 5. The coordinates of the third vertex can not be
(a) \( (6, – 1) \)
(b) \( (4, 5) \)
(c) \( (– 1, 20) \)
(d) \( (2, 9) \)
Answer: (d) \( (2, 9) \)

Question. If \( A(at^2, 2at), B(a/t^2, – 2a/t) \) and \( S (a, 0) \) are three points, then \( \frac{1}{SA} + \frac{1}{SB} \) is independent of
(a) a
(b) t
(c) both a and t
(d) none of these
Answer: (b) t

Question. One diagonal of a square is the portion of the line \( 3x + 2y = 12 \) intercepted between the axes. The coordinates of the extremity of the other diagonals not lying in the first quadrant are
(a) \( (1, – 1) \)
(b) \( (– 1, – 1) \)
(c) \( (– 1, 1) \)
(d) none of these
Answer: (c) \( (– 1, 1) \)

Question. ABCD is a square in which A lies on the positive y-axis and B lies on the positive x-axis. If D is the point (12, 17), the coordinates of C are
(a) (17 , 12)
(b) (17 , 5)
(c) (14 , 16)
(d) (15, 3)
Answer: (b) (17 , 5)

Question. ABCD is a rhombus. Its diagonals AC and BD intersect at the point M and satisfy \( BD = 2AC \). If the coordinates of D and M are (1, 1) and \( (2, – 1) \) respectively, the coordinates of A are
(a) \( (– 3, – 1/2) \)
(b) \( (1, – 3/2) \)
(c) \( (3/2, – 1) \)
(d) \( (1/2, – 3) \)
Answer: (b) \( (1, – 3/2) \)

Question. The points \( (0, 8/3), (1, 3) \) and \( (82, 30) \) are the vertices of
(a) obtuse angled triangle
(b) acute angled triangle
(c) right angled triangle
(d) none of these
Answer: (d) none of these

Question. Area of the rhombus enclosed by the lines \( ax \pm by \pm c = 0 \) is
(a) \( 2a^2/bc \)
(b) \( 2b^2/ca \)
(c) \( 2c^2/ab \)
(d) None
Answer: (c) \( 2c^2/ab \)

Question. The coordinates of the points A and B are respectively \( (– 3, 2) \) and \( (2, 3) \). P and Q are points on the line joining A and B such that \( AP = PQ = QB \). A square PQRS is constructed on PQ as one side, the coordinates of R are
(a) \( (–4/3, 7/3) \)
(b) \( (0, 13/3) \)
(c) \( (1/3, 8/3) \)
(d) \( (2/3, 1) \)
Answer: (d) \( (2/3, 1) \)

Question. A ray of light coming from the point (1, 2) is reflected at a point A on the axis of x and then passes through the point (5, 3). The coordinates of the point A are
(a) (5/13, 0)
(b) \( (–7, 0) \)
(c) (13/5, 0)
(d) (15, 0)
Answer: (c) (13/5, 0)

Question. The centroid of a triangle lies at the origin and the coordinates of its two vertices are \( (– 8, 0) \) and (9, 11), the area of the triangle in sq. units is
(a) 11 / 8
(b) 8/11
(c) 88
(d) None
Answer: (d) None

Question. The line \( 3x + 2y = 24 \) meets the y-axis at A and the x-axis at B ; C is a point on the perpendicular bisector of AB such that the area of the triangle ABC is 91 sq. units. The coordinates of C can be
(a) \( (29/2, – 1) \)
(b) \( (29/2, 13) \)
(c) \( (– 13/2, 1) \)
(d) \( (– 13/2, 13) \)
Answer: (b) \( (29/2, 13) \)

Question. If the straight lines \( x + 2y – 9 = 0, 3x + 5y – 5 = 0 \) and \( ax + by + 1 = 0 \) are concurrent, then the straight line \( 35x – 22y – 1 = 0 \) passes through
(a) (a, b)
(b) (b, a)
(c) (a, – b)
(d) (– a, b)
Answer: (a) (a, b)

Question. Line formed by joining \( (–1, 1) \) and (5,7) is divided by a line \( x + y = 4 \) in the ratio of
(a) 1 : 2
(b) 1 : 3
(c) 3 : 4
(d) 1 : 4
Answer: (a) 1 : 2

Question. The area of triangle formed by the points \( (p, 2 – 2p), (1 – p, 2p) \) and \( (–4 – p, 6 – 2p) \) is 70 sq. units. How many integral value of p are possible ?
(a) 2
(b) 3
(c) 4
(d) None of these
Answer: (d) None of these

Question. Which of the following is TRUE regarding the graphs of the equations of a linear quadratic system ?
(a) the graphs may intersect in two locations
(b) the graphs may intersect in one location
(c) the graphs may not intersect
(d) all three choice are true
Answer: (d) all three choice are true

Question. The midpoints of sides of a triangle are (3, 4), (4, 1) and (2, 0). Which of the following does not denote the co-ordinates of its vertices ?
(a) 1, 3
(b) 5, 3
(c) 5, 5
(d) 3, –3
Answer: (b) 5, 3

Question. The point which lies on the perpendicular bisector of the line sequence joining the points \( P(–2, 0) \) and Q(2, 5) is :-
(a) (0, 0)
(b) (0, 2)
(c) (2, 0)
(d) \( (–2, 0) \)
Answer: (a) (0, 0)

Question. In what ratio is the line segment joining the point \( (–2, –3) \) and (3, 7) divided by y-axis ?
(a) 3 : 2
(b) 5 : 2
(c) 2 : 3
(d) 2 : 5
Answer: (c) 2 : 3

Question. If points (a, 0), (0, b) and (1, 1) are collinear, then \( \frac{a+b}{ab} \) equals :-
(a) 1
(b) –1
(c) 2
(d) \( \sqrt{2} \)
Answer: (a) 1

Question. The point which divides the line segment joining the points (3, 4) and \( (7, –6) \) in the ratio 1 : 2 internally lies in the quadrant.
(a) I
(b) II
(c) III
(d) IV
Answer: (d) IV

Question. The point (22, 23) divide the join of P(7, 5) and Q externally in the ratio 3 : 5, then Q =
(a) (3, 7)
(b) \( (–3, 7) \)
(c) \( (3, –7) \)
(d) \( (–3, –7) \)
Answer: (d) \( (–3, –7) \)

Question. The centroid of a triangle formed by (7, p), (q, –6), (9, 10) is (6, 3), then (p, q) =
(a) (4, 5)
(b) (5, 4)
(c) \( (–5, –2) \)
(d) (5, 2)
Answer: (d) (5, 2)

Question. Mid-points of the sides AB and AC of \(\triangle ABC\) are (3, 5) and \( (–3, –3) \) respectively, then the length of BC =
(a) 10
(b) 15
(c) 20
(d) 30
Answer: (c) 20

Question. ABC is an isosceles triangle with B = (1, 3) and C = \( (–2, 7) \) then A =
(a) (5/6, 6)
(b) (6, 5/6)
(c) (7, 1/8)
(d) None of these
Answer: (a) (5/6, 6)

Question. The third vertex of an equilateral triangle whose two vertices are (2, 4), (2, 6) is :
(a) \( ( \sqrt{3}, 5) \)
(b) \( (2 \sqrt{3}, 5) \)
(c) \( (2+ \sqrt{3}, 5) \)
(d) (2, 5)
Answer: (c) \( (2+ \sqrt{3}, 5) \)

Question. If the points (x, 4) lies on a circle whose centre is at the origin and radius is 5, then x=
(a) \( \pm 5 \)
(b) \( \pm 3 \)
(c) 0
(d) \( \pm 4 \)
Answer: (b) \( \pm 3 \)

Question. The ratio in which the line segment joining P \( (x_1, –y_1) \) and Q \( (x_2, y_2) \) is divided by x-axis is:
(a) \( y_1 : y_2 \)
(b) \( –y_1 : y_2 \)
(c) \( x_1 : x_2 \)
(d) \( –x_1 : x_2 \)
Answer: (b) \( –y_1 : y_2 \)

Question. If the line segment joining the points (3,–4) and (1, 2) is trisected at points P (a,–2) and \( Q \left( \frac{5}{3}, b \right) \), then
(a) \( a = \frac{8}{3}, b = \frac{2}{3} \)
(b) \( a = \frac{7}{3}, b = 0 \)
(c) \( a = \frac{1}{3}, b = 1 \)
(d) \( a = \frac{2}{3}, b = \frac{1}{3} \)
Answer: (b) \( a = \frac{7}{3}, b = 0 \)

Question. If the coordinates of two points A and B are (3, 4) and \( (5, –2) \) respectively. Find the coordinates of any points P, if PA=PB and area of \(\triangle PAB=10\).
(a) (7, 1) or (0, 1)
(b) (7, 2) or (1, 0)
(c) (2, 7) or (1, 0)
(d) (2, 7) or (0, 1)
Answer: (b) (7, 2) or (1, 0)

Question. Find the area of the quadrilateral whose vertices , taken in order are \( (–4, –2,) (–3, –5) (3,–2) \) and (2, 3).
(a) 20 sq. unit
(b) 28 sq. unit
(c) 30 sq. unit
(d) 38 sq. unit
Answer: (b) 28 sq. unit

Question. The area of a triangle is 5. Two of its vertices are (2, 1) and \( (3, –2) \). The third vertex is (x, y) where \( y = x+3 \). Find the coordinates of the third vertex.
(a) \( \left( \frac{13}{2}, \frac{7}{2} \right) \text{ or } \left( \frac{–3}{2}, \frac{3}{2} \right) \)
(b) \( \left( \frac{7}{2}, \frac{13}{2} \right) \text{ or } \left( \frac{3}{2}, \frac{–3}{2} \right) \)
(c) \( \left( \frac{2}{7}, \frac{2}{13} \right) \text{ or } \left( \frac{2}{3}, \frac{–2}{3} \right) \)
(d) \( \left( \frac{7}{2}, \frac{13}{2} \right) \text{ or } \left( \frac{–3}{2}, \frac{3}{2} \right) \)
Answer: (d) \( \left( \frac{7}{2}, \frac{13}{2} \right) \text{ or } \left( \frac{–3}{2}, \frac{3}{2} \right) \)

Question. Find the area of a rhombus, if its vertices are (3, 0) (4, 5), \( (–1, 4) \) and \( (–2, –1) \), taken is order.
(a) 21
(b) 22
(c) 24
(d) 23
Answer: (c) 24

Question. Point P divides the line segment joining the points A (2, 1) and \( (5, –8) \) such that \( \frac{AP}{AB} = \frac{1}{3} \). If P lies on the line \( 2x–y+k=0 \), find the value of k.
(a) 2
(b) –2
(c) 4
(d) –4
Answer: (d) –4

Question. What type of triangle is formed by the points \( P( \sqrt{2}, \sqrt{2}), Q (– \sqrt{2}, – \sqrt{2}) \) and \( R(– \sqrt{6}, \sqrt{6}) \) ?
(a) Scalene triangle
(b) equilateral triangle
(c) Isosceles triangle
(d) None of these
Answer: (b) equilateral triangle

Question. The centre of a circle is \( (2a, a–7) \). Find the values of a, if the circle passes through points \( (11,–9) \) and has diameter \( 10 \sqrt{2} \) units
(a) 2 or 3
(b) 3 or 4
(c) 4 or 2
(d) 3 or 5
Answer: (d) 3 or 5

Question. If \( (–2, 1) \) is centroid of the triangle having its vertices at \( (x, 0), (5,–2), (–8, y) \), then x,y satisfy the relation
(a) \( 5x+3y=0 \)
(b) \( 3x–8y=0 \)
(c) \( 8x+3y=0 \)
(d) \( 8x=3y \)
Answer: (a) \( 5x+3y=0 \)

Question. The point which divides the line segment joining the points (7, –6) and (3, 4) in the ratio 1 : 2 internally lies in the :
(a) I quadrant
(b) II quadrant
(c) III quadrant
(d) IV quadrant
Answer: (d) IV quadrant

Question. A postman walks towards North a distance of 120 m to deliver a letter. He then goes towards East for a distance of 50 m for delivering another letter. The shortest distance between the two places is
(a) 70 m
(b) 120 m
(c) 130 m
(d) 170 m
Answer: (c) 130 m