Coordinate Geometry Class 10 Mathematics Concept notes and Formula Sheet

 

Concepts:

1. Introduction to Coordinate Geometry:

   • Coordinate Geometry is a branch of mathematics that uses algebraic equations to describe the position of points on a plane.
   • The Cartesian plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).
   • The point of intersection of the x-axis and y-axis is called the origin (0, 0).

2. Coordinates of a Point:

   • Any point on the plane is represented by an ordered pair $(x, y)$, where $x$ is the abscissa (distance from the y-axis) and $y$ is the ordinate (distance from the x-axis).
   • The coordinates of the origin are (0, 0).

3. Distance Formula:

   • The distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is given by: $$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
   • This formula is derived from the Pythagorean theorem.

4. Section Formula:

   • The coordinates of the point $P(x, y)$ that divides the line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$ are given by: $$x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n}$$
   • If the point divides the line segment internally, use the above formula. If it divides externally, change the sign in the denominator to $m-n$.

5. Midpoint Formula:

   • The coordinates of the midpoint $M(x, y)$ of the line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are: $$x = \frac{x_1 + x_2}{2}, \quad y = \frac{y_1 + y_2}{2}$$

6. Area of a Triangle:

   • The area of a triangle formed by three points $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ is given by: $$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$
   • The area is positive regardless of the order of the points, but if the area is zero, the points are collinear.

7. Slope of a Line:

   • The slope (m) of a line passing through two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$
   • The slope indicates the steepness and direction of the line. A positive slope means the line is rising, while a negative slope means it is falling.

8. Equation of a Line:

   • The general equation of a line in the Cartesian plane is $Ax + By + C = 0$.
   • The slope-intercept form of a line is: $$y = mx + c$$ where $m$ is the slope and $c$ is the y-intercept.
   • The point-slope form of a line passing through a point $P(x_1, y_1)$ with slope $m$ is: $$y - y_1 = m(x - x_1)$$

9. Collinearity of Points:

   • Three points $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ are collinear if the area of the triangle formed by these points is zero. Alternatively, the slope between any two pairs of these points should be equal.

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Formula Sheet:

1. Distance Formula:

   $$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
   • Used to calculate the distance between two points.

2. Section Formula (Internal Division):

   $$\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$$
   • Used to find the coordinates of a point dividing a line segment internally.

3. Midpoint Formula:

   $$M\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$
   • Used to find the midpoint of a line segment.

4. Area of a Triangle:

   $$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$
   • Used to calculate the area of a triangle formed by three points.

5. Slope of a Line:

   $$m = \frac{y_2 - y_1}{x_2 - x_1}$$
   • Used to determine the slope of a line.

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

   $$y = mx + c$$
   • Used to express a line's equation when the slope and y-intercept are known.

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

   $$y - y_1 = m(x - x_1)$$
   • Used to express a line's equation when the slope and a point on the line are known.

8. Collinearity Condition:

   • Points $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ are collinear if: $$\text{Area of } \triangle ABC = 0$$ or $$\text{Slope of } AB = \text{Slope of } BC$$