Class 10 Mathematics — Chapter 7: Coordinate Geometry

Objective Question Bank

A. MCQs

1. The coordinates of a point on the x-axis are:
   A) (0,y) B) (x,0) C) (x,x) D) (0,0)
2. The coordinates of a point on the y-axis are:
   A) (x,0) B) (x,y) C) (0,y) D) (y,y)
3. The distance between P(x₁,y₁) and Q(x₂,y₂) is:
   A) √[(x₂−x₁)+(y₂−y₁)] B) √[(x₂−x₁)²+(y₂−y₁)²] C) (x₂−x₁)²+(y₂−y₁)² D) √(x₁²+y₁²)
4. The distance of P(x,y) from the origin is:
   A) x+y B) √(x+y) C) √(x²+y²) D) x²+y²
5. Distance between (4,0) and (6,0):
   A) 2 B) 4 C) 6 D) 10
6. Distance between (0,3) and (0,8):
   A) 3 B) 5 C) 8 D) 11
7. Distance between (0,0) and (3,4):
   A) 3 B) 4 C) 5 D) 7
8. The distance formula is based on:
   A) Basic Proportionality Theorem B) Pythagoras theorem C) Midpoint theorem D) Euclid’s division lemma
9. If A,B,C are collinear and B lies between A,C, then:
   A) AB−BC=AC B) AB+BC=AC C) AB×BC=AC D) AB=BC=AC
10. If a²+b²=c², the triangle is:
    A) Equilateral B) Isosceles C) Right-angled D) Obtuse
11. A quadrilateral with four equal sides and one 90° angle is a:
    A) Rectangle B) Rhombus C) Square D) Parallelogram
12. If P is equidistant from A and B:
    A) PA+PB=0 B) PA=PB C) PA>PB D) PA<PB
13. The locus of points equidistant from two fixed points is their:
    A) Angle bisector B) Perpendicular bisector C) x-axis D) y-axis
14. For an equidistant point P, a convenient condition is:
    A) PA+PB=0 B) PA²=PB² C) PA×PB=1 D) PA=0
15. In the section formula, the x-coordinate of the point dividing AB in m₁:m₂ is:
    A) (m₁x₁+m₂x₂)/(m₁+m₂)
    B) (m₁x₂+m₂x₁)/(m₁+m₂)
    C) (x₁+x₂)/2
    D) m₁x₂+m₂x₁
16. The section formula finds:
    A) Distance B) Coordinates of a point dividing a segment in a given ratio C) Slope D) Area
17. A midpoint divides a segment in:
    A) 1:2 B) 2:1 C) 1:1 D) 3:1
18. Midpoint of (x₁,y₁),(x₂,y₂) is:
    A) ((x₁+x₂)/2,(y₁+y₂)/2)
    B) (x₁+x₂,y₁+y₂)
    C) ((x₁−x₂)/2,(y₁−y₂)/2)
    D) (x₁x₂,y₁y₂)
19. If P divides AB internally in 1:2, P is:
    A) Midpoint B) Closer to A C) Closer to B D) Outside AB
20. If P divides AB internally in 2:1, P is:
    A) Closer to A B) Closer to B C) Midpoint D) Outside AB
21. The point dividing (4,−3) and (8,5) in ratio 3:1 is:
    A) (5,1) B) (7,3) C) (6,2) D) (8,5)
22. The trisection points of (2,−2) and (−7,4) are:
    A) (−1,0),(−4,2) B) (1,0),(4,2) C) (−2,1),(−4,2) D) (−1,2),(−4,0)
23. If P lies on the y-axis, its x-coordinate is:
    A) 1 B) −1 C) 0 D) y
24. If P lies on the x-axis, its y-coordinate is:
    A) 0 B) 1 C) x D) −1
25. The diagonals of a parallelogram:
    A) Are always equal B) Are perpendicular C) Bisect each other D) Are parallel
26. One way to prove four points form a square is to show:
    A) One pair of equal sides
    B) All four sides equal and both diagonals equal
    C) Only diagonals perpendicular
    D) All coordinates positive
27. For a point equidistant from (7,1) and (3,5), the relation obtained is:
    A) x+y=2 B) x−y=2 C) x+y=4 D) x−y=4
28. The point on the y-axis equidistant from (6,5) and (−4,3) is:
    A) (0,9) B) (9,0) C) (0,−9) D) (6,9)
29. (−4,6) divides A(−6,10), B(3,−8) internally in:
    A) 1:2 B) 2:7 C) 7:2 D) 3:7
30. The y-axis divides the segment joining (5,−6) and (−1,−4) in:
    A) 1:5 B) 5:1 C) 2:5 D) 5:2

B. Fill in the Blanks

1. The x-coordinate is called the ________.
2. The y-coordinate is called the ________.
3. A point on the x-axis has its ________ coordinate zero.
4. A point on the y-axis has its ________ coordinate zero.
5. The coordinates of the origin are ________.
6. The distance formula is based on the ________ theorem.
7. Distance is always ________.
8. Distance of (x,y) from the origin is ________.
9. Points lying on the same straight line are called ________ points.
10. Points equidistant from two fixed points lie on their ________.
11. The section formula gives coordinates of a point dividing a segment in a given ________.
12. The midpoint divides a segment in the ratio ________.
13. The x-coordinate of the midpoint is ________.
14. The y-coordinate of the midpoint is ________.
15. The diagonals of a parallelogram ________ each other.
16. A point outside a segment but on its line divides it ________.
17. In internal division, the dividing point lies ________ the endpoints.
18. Trisection divides a segment into ________ equal parts.

C. True / False

1. Every point on the x-axis has y-coordinate zero.
2. Every point on the y-axis has x-coordinate zero.
3. Distance between two points can be negative.
4. Distance formula is derived using Pythagoras theorem.
5. Midpoint divides a segment in 1:2.
6. Midpoint divides a segment in 1:1.
7. Three collinear points form a triangle.
8. If B lies between A,C and they are collinear, AB+BC=AC.
9. A point equidistant from two fixed points lies on their perpendicular bisector.
10. The section formula in this chapter is for internal division.
11. Diagonals of a parallelogram bisect each other.
12. Midpoint coordinates are obtained by subtracting endpoint coordinates.
13. Four equal sides and one right angle give a square.
14. Trisection points divide a segment into two equal parts.
15. The x-coordinate is called the ordinate.

D. Assertion–Reason

Choose:
A) Both true; Reason correctly explains Assertion
B) Both true; Reason does not explain Assertion
C) Assertion true; Reason false
D) Assertion false; Reason true

1. Assertion: Distance between two points is never negative.
   Reason: Distance represents a length.
2. Assertion: Distance between (0,0) and (3,4) is 5 units.
   Reason: Distance formula is based on Pythagoras theorem.
3. Assertion: Every point on the y-axis has x-coordinate zero.
   Reason: Its horizontal distance from the y-axis is zero.
4. Assertion: The midpoint divides a segment in 1:1.
   Reason: It is equally distant from both endpoints.
5. Assertion: Diagonals of a parallelogram can help find an unknown vertex coordinate.
   Reason: Its diagonals bisect each other.
6. Assertion: If PA=PB, P lies on the perpendicular bisector of AB.
   Reason: Every point on the perpendicular bisector is equidistant from A and B.
7. Assertion: The section formula finds distance between two points.
   Reason: It gives coordinates of a point dividing a segment in a given ratio.
8. Assertion: Three points can be tested for collinearity using the distance formula.
   Reason: For suitable ordering, two adjacent distances add to the third.

E. Very Short Answer

1. What is the abscissa of a point?
2. What is the ordinate of a point?
3. Write the coordinates of the origin.
4. Write the general form of a point on the x-axis.
5. Write the general form of a point on the y-axis.
6. Write the distance formula.
7. Write the distance of P(x,y) from the origin.
8. In what ratio does a midpoint divide a segment?
9. Write the midpoint formula.
10. What are collinear points?
11. What condition shows P is equidistant from A and B?
12. What is the locus of points equidistant from two fixed points?
13. State the useful property of parallelogram diagonals.
14. What does the section formula determine?
15. What are trisection points?

F. Calculation-Based Objective Questions

1. Distance between (2,3) and (4,1):
   A) √2 B) 2√2 C) 4 D) 8
2. Distance between (−5,7) and (−1,3):
   A) 4 B) 4√2 C) 8 D) √8
3. Distance between (0,0) and (36,15):
   A) 39 B) 51 C) 21 D) 36
4. If P(2,−3), Q(10,y) are 10 units apart, y can be:
   A) 3 or −9 B) 3 or −3 C) 9 or −3 D) 7 or −1
5. Midpoint of (2,4) and (6,8):
   A) (4,6) B) (8,12) C) (2,2) D) (3,4)
6. Point dividing (−1,7) and (4,−3) in ratio 2:3:
   A) (1,3) B) (2,1) C) (1,1) D) (3,1)
7. If A(6,1), B(8,2), C(9,4), D(p,3) are consecutive vertices of a parallelogram, p =
   A) 5 B) 6 C) 7 D) 8
8. Midpoint of (−2,−2) and (2,−4):
   A) (0,−3) B) (0,3) C) (2,−3) D) (−2,0)

G. Concept-Trap Questions

1. A student gives the midpoint of (2,6) and (8,4) as (10,10). Is it correct?
2. Can two distinct points have distance zero?
3. On which axis does (0,−7) lie?
4. On which axis does (−5,0) lie?
5. If AB=5, BC=7 and AC=12, with B between A,C, are A,B,C collinear?
6. A triangle has sides 5,12,13. What type of triangle is it?
7. If a point divides AB in 1:1, which formula should be used?
8. If a point lies on the y-axis while using section formula, which coordinate is zero?
9. If a point lies on the x-axis, which coordinate is zero?
10. Which parallelogram property is useful for finding an unknown vertex?

ANSWER KEY

A. MCQs

1-B, 2-C, 3-B, 4-C, 5-A, 6-B, 7-C, 8-B, 9-B, 10-C,
11-C, 12-B, 13-B, 14-B, 15-B, 16-B, 17-C, 18-A, 19-B, 20-A,
21-B, 22-A, 23-C, 24-A, 25-C, 26-B, 27-B, 28-A, 29-B, 30-B.

B. Fill in the Blanks

1. abscissa
2. ordinate
3. y
4. x
5. (0,0)
6. Pythagoras
7. non-negative
8. √(x²+y²)
9. collinear
10. perpendicular bisector
11. ratio
12. 1:1
13. (x₁+x₂)/2
14. (y₁+y₂)/2
15. bisect
16. externally
17. between
18. three

C. True / False

1-T, 2-T, 3-F, 4-T, 5-F, 6-T, 7-F, 8-T, 9-T, 10-T,
11-T, 12-F, 13-T, 14-F, 15-F.

D. Assertion–Reason

1-A, 2-A, 3-A, 4-A, 5-A, 6-A, 7-D, 8-A.

E. Very Short Answer

1. x-coordinate
2. y-coordinate
3. (0,0)
4. (x,0)
5. (0,y)
6. √[(x₂−x₁)²+(y₂−y₁)²]
7. √(x²+y²)
8. 1:1
9. ((x₁+x₂)/2,(y₁+y₂)/2)
10. Points lying on the same straight line
11. PA=PB, or PA²=PB²
12. Perpendicular bisector
13. They bisect each other
14. Coordinates of a point dividing a segment in a given ratio
15. Points dividing a segment into three equal parts

F. Calculation-Based

1-B, 2-B, 3-A, 4-A, 5-A, 6-C, 7-C, 8-A.

G. Concept-Trap

1. No; midpoint = (5,5).
2. No.
3. y-axis.
4. x-axis.
5. Yes.
6. Right-angled triangle.
7. Midpoint formula.
8. x-coordinate.
9. y-coordinate.
10. Diagonals of a parallelogram bisect each other.