Quick Answer – What is Coordinate Geometry?
Coordinate geometry (or analytic geometry) uses a coordinate system to represent geometric figures and derive relationships between points. It provides algebraic formulas for distance, midpoint, section formula, area of triangle, centroid, and slope of a line, forming the backbone of JEE and NCERT mathematics.
Key Formulas
Distance: d = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Section (internal): P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
Area of Δ: ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Centroid: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
Slope (m) = (y₂−y₁)/(x₂−x₁)
Solved Examples
Example 1: Distance between (3,4) and (7,1)
d = √[(7−3)² + (1−4)²] = √(16+9) = √25 = 5
Example 2: Area of triangle (1,2), (4,3), (2,5)
Area = ½ |1(3−5) + 4(5−2) + 2(2−3)| = ½ |−2+12−2| = ½ × 8 = 4 sq. units
Example 3: Section formula (internal) A(2,3), B(5,7) ratio 1:2
x = (1×5 + 2×2)/(1+2) = (5+4)/3 = 3
y = (1×7 + 2×3)/3 = (7+6)/3 = 13/3 ≈ 4.333 → P(3, 4.333)
Frequently Asked Questions
Q1. Can I use negative coordinates?
Absolutely. Enter negative numbers with a minus sign (e.g., −3). The formulas handle all real numbers correctly.
Q2. What if the denominator is zero in slope or section formula?
If the line is vertical (x₁=x₂), the slope is undefined. The calculator will display 'Undefined'. In the section formula, if m+n=0, it's invalid; the tool warns you.
Q3. Does the area of triangle work for collinear points?
Yes. The area will be 0, which indicates the points are collinear.
Q4. Can I use decimal coordinates?
Yes, decimals like 2.5 or −0.75 are accepted. Commas are automatically converted to dots.
Q5. Is the section formula for internal division only?
This tool covers internal division. For external division, simply use a negative ratio (e.g., m:n with n negative) and the same formula applies; the calculator will handle the sign automatically.