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Coordinate Geometry Class 10 Notes Maths Chapter 7

What is a Coordinate System?

A Coordinate System is a mathematical framework used to determine the position or location of points in space. It provides a way to describe the position of objects or points using numerical values called coordinates.
  • As shown in the figure, the line XOX′ is known as the X-axis, and YOY′ is known as the Y-axis.
  • The point O is called the origin. For any point P (x y), the ordered pair (x,y) is called the coordinate of point P.
  •  The distance of a point from the Y-axis is called its abscissa and the distance of a point from the X-axis is called its ordinate.

Coordinate Geometry Class 10 Notes Maths Chapter 7

Distance Formula

The distance between two points P (x1, y1) and Q (x2, y2) is given by:
Coordinate Geometry Class 10 Notes Maths Chapter 7
This is also known as the Distance Formula.

Coordinate Geometry Class 10 Notes Maths Chapter 7

Note: The distance of any point P(x,y) from the origin O(0,0) is given by:

Coordinate Geometry Class 10 Notes Maths Chapter 7

Example 1: Find the distance between the points D and E, in the given figure.

Coordinate Geometry Class 10 Notes Maths Chapter 7

Solution:

Coordinate Geometry Class 10 Notes Maths Chapter 7 

Example 2: What is the distance between two points (2, 3) and (-4, 5) using the distance formula?

Sol: The distance formula is used to calculate the distance between two points in a coordinate plane. It is given as:

d = √[(x2 - x1)² + (y2 - y1)²]

Using this formula, we can find the distance between the points (2, 3) and (-4, 5) as follows:

d = √[(-4 - 2)² + (5 - 3)²]

d = √[(-6)² + (2)²]

d = √[36 + 4]

d = √40

d = 6.32 (approx.)

Therefore, the distance between the points (2, 3) and (-4, 5) is approximately 6.32 units.

Question for Chapter Notes: Coordinate Geometry
Try yourself:What is the distance between points (3,4) and (-2,1)?
View Solution

Section Formula

Let P (x,y) be a point on the line segment joining A(x1, y1) and B(x2, y2) such that it divides AB internally in the ratio m:n. The coordinates of the point are given by
Coordinate Geometry Class 10 Notes Maths Chapter 7
Coordinate Geometry Class 10 Notes Maths Chapter 7
This is known as the Section Formula.

Coordinate Geometry Class 10 Notes Maths Chapter 7

Note:

(i) If the point P divides the line segment joining A(x1, y1) and B(x2, y2) internally in the ratio k:1, its coordinates are given by:
Coordinate Geometry Class 10 Notes Maths Chapter 7

Example 2: In what ratio does the point (2,- 5) divide the line segment joining the points A(-3, 5) and B(4, -9).

Sol: Let the ratio be λ : 1
Coordinate Geometry Class 10 Notes Maths Chapter 7

We have put m = λ and n = 1
or
Coordinate Geometry Class 10 Notes Maths Chapter 7
But, coordinates of point is given as p(2,-5) 

Coordinate Geometry Class 10 Notes Maths Chapter 7

But, coordinates of point is given as p(2,-5) 

Coordinate Geometry Class 10 Notes Maths Chapter 7

4λ - 3 = 2(λ + 1)
⇒ 4λ = 2λ + 2 + 3

⇒ 2λ = 5
λ = 5/2

The required ratio is 5:2.

Mid -Point Formula

The mid-point of the line joining A(x1, y1) and B(x2, y2) is given as
Coordinate Geometry Class 10 Notes Maths Chapter 7

Example 3: Suppose we have two points A(2, 4) and B(6, 8). We want to find the midpoint of the line segment AB.

Using the midpoint formula:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

= ((2 + 6) / 2, (4 + 8) / 2)

= (8 / 2, 12 / 2)

= (4, 6)

Therefore, the midpoint of the line segment AB is M(4, 6).

Some Solved Questions

Q1: Find the distance between the points (3, 5) and (-2, -1) using the distance formula.

Solution:

Using the distance formula:

d = √[(x2 - x1)² + (y2 - y1)²]

Substituting the coordinates:

d = √[(-2 - 3)² + (-1 - 5)²]

d = √[(-5)² + (-6)²]

d = √[25 + 36]

d = √61

Therefore, the distance between the points (3, 5) and (-2, -1) is √61 units.

Q2: Find the coordinates of the midpoint of the line segment joining the points (-3, 2) and (5, -4).

Solution:  

Using the midpoint formula:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

Substituting the coordinates:

Midpoint = ((-3 + 5) / 2, (2 + (-4)) / 2)

Midpoint = (2 / 2, -2 / 2)

Midpoint = (1, -1)

Therefore, the midpoint of the line segment joining (-3, 2) and (5, -4) is (1, -1).

The document Coordinate Geometry Class 10 Notes Maths Chapter 7 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Coordinate Geometry Class 10 Notes Maths Chapter 7

1. What is the Distance Formula in Coordinate Geometry?
Ans. The Distance Formula in Coordinate Geometry is used to find the distance between two points in a coordinate plane. It is given by the formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
2. How is the Mid-Point Formula used in Coordinate Geometry?
Ans. The Mid-Point Formula in Coordinate Geometry is used to find the coordinates of the midpoint of a line segment when the coordinates of the endpoints are known. It is given by the formula: \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
3. What is the Section Formula in Coordinate Geometry?
Ans. The Section Formula in Coordinate Geometry is used to find the coordinates of a point which divides a line segment into two parts in a given ratio. It is given by the formula: \[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
4. How is the Centroid of a Triangle calculated using Coordinate Geometry?
Ans. The Centroid of a Triangle is the point where the three medians of the triangle intersect. It is calculated by finding the average of the coordinates of the three vertices of the triangle.
5. What is the Condition for Collinearity of Three Points in Coordinate Geometry?
Ans. The Condition for Collinearity of Three Points in Coordinate Geometry is that the slope of the line passing through any two points should be equal to the slope of the line passing through the other two points. This indicates that all three points lie on the same straight line.
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