Coordinate Geometry – Class 10 Mathematics notes, important questions and MCQs

Chapter overview

Coordinate geometry uses (x, y) points to find distances between points and to divide a line segment in a given ratio.

Distance formula

• Distance between (x₁, y₁) and (x₂, y₂) = √[(x₂ − x₁)² + (y₂ − y₁)²].
• Distance of (x, y) from the origin = √(x² + y²).
• Use it to test collinearity, or to identify isosceles/equilateral triangles, squares and rhombuses.

Section formula

• Point dividing the join of A(x₁, y₁) and B(x₂, y₂) internally in m : n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).
• Mid-point = ((x₁ + x₂)/2, (y₁ + y₂)/2).
• Take the ratio as k : 1 when the ratio is unknown.

Quadrants and axes

• Signs: I (+, +), II (−, +), III (−, −), IV (+, −).
• Any point on the x-axis is (x, 0); on the y-axis it is (0, y).

Key terms

Abscissa

the x-coordinate of a point

Ordinate

the y-coordinate of a point

Collinear points

points lying on one straight line

Formulas

• d = √[(x₂ − x₁)² + (y₂ − y₁)²]
• P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))
• Mid-point = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Important questions with answers

Quick revision

• Distance formula and section formula are the two key tools.
• Mid-point is the m = n case.
• x-axis point (x, 0), y-axis point (0, y).

Exam tips

• In the section formula, m goes with the second point (x₂, y₂).
• Points are collinear if the sum of two distances equals the third.

Practice MCQs

1. Distance of (3, 4) from the origin is

   1. 7
   2. 5
   3. 25
   4. 1
   Show answer

   Answer: (b) 5. √(9 + 16) = 5.

2. Mid-point of (2, 4) and (6, 8) is

   1. (3, 5)
   2. (8, 12)
   3. (4, 6)
   4. (2, 2)
   Show answer

   Answer: (c) (4, 6). ((2 + 6)/2, (4 + 8)/2).

3. The point (−3, 5) lies in quadrant

   1. II
   2. I
   3. III
   4. IV
   Show answer

   Answer: (a) II. x negative, y positive.

4. Distance between (1, −3) and (4, 1) is

   1. 25
   2. 7
   3. 5
   4. √7
   Show answer

   Answer: (c) 5. √(9 + 16) = 5.

5. Point dividing (−1, 7) and (4, −3) in ratio 2 : 3 is

   1. (2, 3)
   2. (1, 3)
   3. (1, 2)
   4. (3, 1)
   Show answer

   Answer: (b) (1, 3). x = (8 − 3)/5 = 1, y = (−6 + 21)/5 = 3.

6. If (x, 3) is the mid-point of (2, 5) and (6, 1), then x =

   1. 3
   2. 4
   3. 5
   4. 8
   Show answer

   Answer: (b) 4. x = (2 + 6)/2 = 4.

7. The ratio in which the x-axis divides the join of (2, −3) and (5, 6) is

   1. 2 : 1
   2. 1 : 2
   3. 3 : 2
   4. 1 : 3
   Show answer

   Answer: (b) 1 : 2. y = 0: (6k − 3)/(k + 1) = 0 ⇒ k = 1/2.

8. If A(1, 2), B(4, y), C(x, 6), D(3, 5) are vertices of parallelogram ABCD, then (x, y) is

   1. (6, 6)
   2. (3, 6)
   3. (6, 3)
   4. (2, 3)
   Show answer

   Answer: (c) (6, 3). Diagonals bisect: (1 + x)/2 = 7/2 ⇒ x = 6; (2 + 6)/2 = (y + 5)/2 ⇒ y = 3.