CBSE Class 10 Mathematics Coordinate Geometry Notes Set 12

Coordinate Geometry: Key Formulas

Distance between P(x₁, y₁) and Q(x₂, y₂): \( PQ = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \). The distance of P(x, y) from the origin is \( \\sqrt{x^2+y^2} \).
Section formula (internal division in ratio m : n): \( \\left(\\frac{mx_2 + nx_1}{m+n},\\ \\frac{my_2 + ny_1}{m+n}\\right) \). Midpoint: take m = n.
A line ax + by + c = 0 divides the segment joining P and Q in the ratio k : 1 where k = −(ax₁ + by₁ + c)/(ax₂ + by₂ + c). A negative k means external division. The x-axis divides in the ratio −y₁ : y₂.

Triangles and Quadrilaterals

Area of a triangle: \( \\frac{1}{2}\\left|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)\\right| \). Three points are collinear when this area is zero.
Centroid: \( \\left(\\frac{x_1+x_2+x_3}{3},\\ \\frac{y_1+y_2+y_3}{3}\\right) \). Incentre: \( \\left(\\frac{ax_1+bx_2+cx_3}{a+b+c},\\ \\frac{ay_1+by_2+cy_3}{a+b+c}\\right) \) where a, b, c are the sides opposite the vertices.
In an equilateral triangle the centroid, incentre, circumcentre and orthocentre coincide; otherwise the centroid, circumcentre and orthocentre are collinear.
A parallelogram has diagonals that bisect each other, so the fourth vertex is found from midpoint of one diagonal = midpoint of the other.
A quadrilateral is a rhombus if all sides are equal, a rectangle if opposite sides and the diagonals are equal, and a square if all sides and both diagonals are equal.

Lines and Polar Coordinates

The line ax + by = c cuts the axes at (c/a, 0) and (0, c/b); the triangle with the origin has area c²/(2|ab|).
Lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are parallel if a₁/a₂ = b₁/b₂ ≠ c₁/c₂, coincident if all three ratios are equal, intersecting if a₁/a₂ ≠ b₁/b₂, and perpendicular if a₁a₂ + b₁b₂ = 0.
Polar coordinates (r, θ) convert to Cartesian by x = r cos θ, y = r sin θ; conversely r = √(x² + y²) and tan θ = y/x.