Coordinate Geometry — Study Notes

Overview

Coordinate Geometry forms the bridge between algebra and geometry by representing geometric figures using algebraic equations on the Cartesian plane. In SOF IMO, this topic tests your ability to visualize points, calculate distances, and find positions that divide line segments in given ratios. Expect 2–4 questions from this area, often integrated with concepts like triangles, quadrilaterals, or real-life positioning problems.

Mastery requires fluency in plotting points accurately, applying the distance formula without computational errors, and using the section formula for both internal and external division. Many students lose marks through sign errors or mixing up coordinates—precision is essential. This topic is foundational for higher geometry and appears frequently in Achievers Section problems combined with area calculations or optimization.

Key Concepts

• **Cartesian Plane**: A two-dimensional plane defined by perpendicular x-axis (horizontal) and y-axis (vertical) intersecting at origin O(0,0). Any point is represented as an ordered pair (x, y) where x is the abscissa and y is the ordinate.

• **Quadrants**: The plane divides into four quadrants — I (+,+), II (−,+), III (−,−), IV (+,−). The signs of coordinates determine which quadrant a point lies in. Points on axes belong to no quadrant.

• **Distance Formula**: The distance between two points A(x₁, y₁) and B(x₂, y₂) is given by d = √[(x₂−x₁)² + (y₁−y₂)²]. This comes from the Pythagorean theorem applied to the right triangle formed by the points.

• **Section Formula (Internal Division)**: If point P divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in ratio m:n, then P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).

• **Midpoint Formula**: Special case of section formula when m:n = 1:1. Midpoint M = ((x₁+x₂)/2, (y₁+y₂)/2). This is the arithmetic mean of the coordinates.

• **Section Formula (External Division)**: When P divides AB externally in ratio m:n, P = ((mx₂−nx₁)/(m−n), (my₂−ny₁)/(m−n)). Note the subtraction in both numerator and denominator.

• **Collinearity Test**: Three points A, B, C are collinear if the distance AB + BC = AC (or any cyclic permutation). Alternatively, area of triangle ABC equals zero.

• **Origin and Axes**: Distance from origin to point (x, y) is √(x²+y²). Distance of a point from x-axis is |y| and from y-axis is |x|.

Formulas / Key Facts

1. **Distance between A(x₁, y₁) and B(x₂, y₂)**: d = √[(x₂−x₁)² + (y₂−y₁)²]

2. **Distance from origin to (x, y)**: d = √(x² + y²)

3. **Section formula (internal division, ratio m:n)**: P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))

4. **Section formula (external division, ratio m:n)**: P = ((mx₂−nx₁)/(m−n), (my₂−ny₁)/(m−n))

5. **Midpoint of segment AB**: M = ((x₁+x₂)/2, (y₁+y₂)/2)

6. **Centroid of triangle with vertices A(x₁,y₁), B(x₂,y₂), C(x₃,y₃)**: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)

7. **Distance from point (x,y) to x-axis**: |y| and to y-axis: |x|

8. **Reflection formulas**: Reflection of (x,y) in x-axis is (x,−y); in y-axis is (−x,y); in origin is (−x,−y); in line y=x is (y,x)

Worked Examples

**Example 1: Finding Distance** Find the distance between points A(3, 4) and B(−1, 1).

*Solution:* Using distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²] Here x₁=3, y₁=4, x₂=−1, y₂=1 d = √[(−1−3)² + (1−4)²] d = √[(−4)² + (−3)²] d = √[16 + 9] d = √25 = 5 units

**Example 2: Section Formula Application** Find the coordinates of point P that divides the line segment joining A(2, 3) and B(8, 9) internally in ratio 1:2.

*Solution:* Using internal division formula with m=1, n=2: x-coordinate = (mx₂+nx₁)/(m+n) = (1×8 + 2×2)/(1+2) = (8+4)/3 = 12/3 = 4 y-coordinate = (my₂+ny₁)/(m+n) = (1×9 + 2×3)/(1+2) = (9+6)/3 = 15/3 = 5 Therefore P = (4, 5)

**Example 3: Collinearity Check** Check if points A(1, 2), B(3, 6), and C(4, 8) are collinear.

*Solution:* Find distances AB, BC, and AC: AB = √[(3−1)² + (6−2)²] = √[4+16] = √20 = 2√5 BC = √[(4−3)² + (8−6)²] = √[1+4] = √5 AC = √[(4−1)² + (8−2)²] = √[9+36] = √45 = 3√5 Check if AB + BC = AC: 2√5 + √5 = 3√5 ✓ Since this is true, the points are collinear.

Common Mistakes

**Sign errors in coordinates** → When computing (x₂−x₁)², students often forget that (−4)² = 16, not −16. Always square the entire difference, not individual terms. Write out the subtraction explicitly before squaring.

**Confusing ratio order in section formula** → If P divides AB in ratio 2:3, the point P is closer to A. Use m=2 for B's coordinate and n=3 for A's coordinate in the numerator. Wrong: (2x₁+3x₂)/5; Correct: (2x₂+3x₁)/5.

**Mixing internal and external division** → Internal division uses (m+n) in denominator; external uses (m−n). For external division, also subtract in numerator (mx₂−nx₁), not add. Check if point lies between or outside the segment.

**Forgetting square root in distance formula** → Distance is √[(x₂−x₁)² + (y₂−y₁)²], not just (x₂−x₁)² + (y₂−y₁)². The square root is essential and frequently omitted under exam pressure.

**Midpoint confusion with centroid** → Midpoint applies to two points (divide sum by 2); centroid applies to three points of a triangle (divide sum by 3). Don't mix these formulas.

Quick Reference

• **Distance**: d = √[(x₂−x₁)² + (y₂−y₁)²] — always take square root at the end
• **Midpoint**: Average both coordinates — ((x₁+x₂)/2, (y₁+y₂)/2)
• **Internal division m:n**: Weighted average — numerator has cross-multiplication (mx₂+nx₁), denominator is (m+n)
• **External division m:n**: Use subtraction — (mx₂−nx₁)/(m−n)
• **Quadrant signs**: I(+,+), II(−,+), III(−,−), IV(+,−)
• **Collinear test**: Sum of two distances equals third distance along a line