Coordinate Geometry

Learning Objectives

• Calculate the distance between two points using the distance formula: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Apply the section formula to find the coordinates of a point dividing a line segment internally in a given ratio: $x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \; y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2}$.
• Determine the midpoint of a line segment using the midpoint formula: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
• Assess the collinearity of three points by verifying if the sum of the distances between two pairs equals the distance between the third pair.
• Find the points of trisection of a line segment using the section formula.
• Establish a relation between $x$ and $y$ for a point $(x, y)$ to be equidistant from two given points, resulting in a linear equation.
• Utilize the property that diagonals of a parallelogram bisect each other to find unknown coordinates.

Chapter Notes

Distance Formula

• Formula: The distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ in a Cartesian plane is given by: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Example: For points $A(4, 0)$ and $B(6, 0)$, the distance $AB = 6 - 4 = 2$ units.

Section Formula

• Formula: To find the coordinates of a point $P(x, y)$ that divides a line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $m_1:m_2$: $x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \quad y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2}$
• Example: For a point dividing the segment joining $(4, -3)$ and $(8, 5)$ in the ratio $3:1$, the coordinates are $(7, 3)$.

Midpoint Formula

• Formula: The midpoint of a line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Collinearity of Points

• Condition: Three points are collinear if the sum of the distances between two pairs of points equals the distance between the third pair.
• Example: Points $A(3, 1)$, $B(6, 4)$, and $C(8, 6)$ are collinear if $AB + BC = AC$.

Trisection of a Line Segment

• Concept: To find points that divide a line segment into three equal parts, use the section formula for ratios $1:2$ and $2:1$.
• Example: For segment joining $A(2, -2)$ and $B(-7, 4)$, trisection points are $(-1, 0)$ and $(-4, 2)$.

Equidistant Points

• Condition: A point $(x, y)$ is equidistant from two given points if it satisfies the equation derived from setting the distances equal.
• Example: For points $(7, 1)$ and $(3, 5)$, the relation is $x - y = 2$.

Parallelogram Diagonal Bisection

• Property: Diagonals of a parallelogram bisect each other.
• Example: For vertices $A(6, 1)$, $B(8, 2)$, $C(9, 4)$, and $D(p, 3)$, the midpoint condition gives $p = 7$.