Coordinate GeometrY CLASS 10

"The Mysterious Treasure Map: A Journey in Coordinate Geometry"

Once upon a time in a quiet Indian village, there lived two curious friends, Raj and Priya. One sunny afternoon, they stumbled upon an old, weathered treasure map. The map promised hidden riches beyond their wildest dreams but was filled with cryptic clues.

Chapter 1: The Enigmatic Clues

The treasure map's clues seemed perplexing at first. It mentioned coordinates like (3, 4) and (7, 9), leaving Raj and Priya puzzled. Determined to uncover the treasure, they realized they needed to understand coordinate geometry.

Chapter 2: The Cartesian Plane

With the guidance of their elder sister, Meera, Raj and Priya learned about the Cartesian plane. Meera explained that it was like a giant grid covering the Earth, with two perpendicular lines - the x-axis and y-axis. Points were defined by their (x, y) coordinates, just like locations on a map.

Chapter 3: Distance Formula

As Raj and Priya examined the map, they noticed a hidden formula for calculating the distance between two points. Meera showed them the distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2). It allowed them to find the distance between any two locations on the map.

Chapter 4: The Section Formula

The treasure map contained a riddle about a secret passage. Meera introduced them to the section formula for dividing a line segment in a given ratio. It was like cutting a cake into equal parts but for lines. The formula was ((x1m + x2n)/(m + n), (y1m + y2n)/(m + n)).

Chapter 5: The Treasure Unveiled

Armed with their newfound knowledge, Raj and Priya cracked the map's clues. They used the distance formula to measure the precise distance between landmarks and applied the section formula to find the secret passage. It led them to a hidden chest filled with treasure.

The story of Raj and Priya's adventure illustrated the practical applications of coordinate geometry, including the distance formula and section formula. They learned that math was not just about numbers but could unlock hidden secrets and treasures in the real world.

1. Distance Formula:

Formula: d = √((x2 - x1)^2 + (y2 - y1)^2)

Used to find the distance between two points (x1, y1) and (x2, y2) in a Cartesian plane.

2. Midpoint Formula:

Formula: M = ((x1 + x2)/2, (y1 + y2)/2)

Calculates the midpoint between two points (x1, y1) and (x2, y2).

3. Slope Formula:

Formula: m = (y2 - y1)/(x2 - x1)

Gives the slope of a line passing through two points (x1, y1) and (x2, y2).

4. Equation of a Line (Point-Slope Form):

Formula: y - y1 = m(x - x1)

Represents the equation of a line with slope m passing through a point (x1, y1).

5. Equation of a Line (Slope-Intercept Form):

Formula: y = mx + b

Describes a line's equation, where m is the slope and b is the y-intercept.

6. Equation of a Circle:

Formula: (x - h)^2 + (y - k)^2 = r^2

Represents the equation of a circle centered at (h, k) with radius r.

7. Equation of an Ellipse (Standard Form):

Formula: (x - h)^2/a^2 + (y - k)^2/b^2 = 1

Describes the equation of an ellipse centered at (h, k) with semi-major axis a and semi-minor axis b.

8. Equation of a Parabola (Vertex Form):

Formula: y = a(x - h)^2 + k

Represents a parabola with vertex (h, k) and vertical axis of symmetry.

9. Equation of a Hyperbola (Standard Form):

Formula: (x - h)^2/a^2 - (y - k)^2/b^2 = 1

Describes the equation of a hyperbola centered at (h, k) with horizontal transverse axis.

10. Parallel and Perpendicular Lines:

- For parallel lines, slopes are equal: m1 = m2.

- For perpendicular lines, slopes are negative reciprocals: m1 * m2 = -1.

These are some of the fundamental formulas and concepts in coordinate geometry. They are essential for solving problems involving points, lines, circles, and other geometric shapes in a Cartesian coordinate system.