Overview of Rectangular Cartesian Coordinates in JEE Maths

Rectangular Cartesian Coordinates form the foundation of coordinate geometry in JEE Maths. Introduced by René Descartes, this system helps represent points on a plane using ordered pairs (x, y). Mastering Cartesian coordinates is essential for solving lines, distance, midpoint, slope, and geometric problems efficiently. At StudentBro.in, we provide a comprehensive guide covering formulas, properties, examples, and practice problems to help JEE aspirants excel in coordinate geometry.

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What are Rectangular Cartesian Coordinates?

The rectangular Cartesian coordinate system consists of:

• x-axis – horizontal axis

• y-axis – vertical axis

• Origin (0, 0) – intersection of x and y axes

• Point (x, y) – represents a location on the plane

Example: Point P(3, 4) is 3 units along x-axis and 4 units along y-axis.

Distance Formula

The distance between two points P(x₁, y₁) and Q(x₂, y₂) is:

D = √[(x₂ - x₁)² + (y₂ - y₁)²]

Example: Distance between A(1, 2) and B(4, 6) → √[(4-1)² + (6-2)²] = √(9+16) = 5

Line

Midpoint Formula

The midpoint of line segment joining P(x₁, y₁) and Q(x₂, y₂) is:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Example: Midpoint of A(2, 3) and B(4, 7) → M = ((2+4)/2, (3+7)/2) = (3, 5)

Slope of a Line

The slope (m) of a line through points P(x₁, y₁) and Q(x₂, y₂) is:

m = (y₂ - y₁) / (x₂ - x₁)

• Positive slope: Line rises from left to right

• Negative slope: Line falls from left to right

• Zero slope: Horizontal line

• Undefined slope: Vertical line

Example: Slope of line joining (1, 2) and (3, 6) → m = (6-2)/(3-1) = 2

Equation of a Line in Rectangular Coordinates

Common forms:

1. Slope-Intercept Form: y = mx + c

2. Point-Slope Form: y - y₁ = m(x - x₁)

3. Two-Point Form: (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)

4. General Form: Ax + By + C = 0

Example: Line passing through (2, 3) with slope 2 → y - 3 = 2(x - 2) → y = 2x - 1

Area of a Triangle Using Coordinates

If vertices of a triangle are A(x₁, y₁), B(x₂, y₂), C(x₃, y₃):

Area = 1/2 | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |

Example: A(0,0), B(4,0), C(0,3) → Area = 1/2 |0(0-3) + 4(3-0) + 0(0-0)| = 6

Distance of a Point from a Line

Line in general form: Ax + By + C = 0
Point P(x₁, y₁)

Distance = |Ax₁ + By₁ + C| / √(A² + B²)

Example: Distance of (1, 2) from line 3x + 4y - 5 = 0 → |31 + 42 - 5| / √(9+16) = 6/5

Properties and Applications

• Calculate length, slope, and midpoint efficiently

• Solve lines intersection and parallel/perpendicular lines problems

• Find areas using coordinates

• Solve JEE Main & Advanced coordinate geometry questions

• Applicable in mechanics, physics, and analytic geometry problems

Tips and Tricks for JEE Rectangular Cartesian Coordinates

1. Memorize distance, midpoint, slope, and area formulas

2. Check special points like origin, x-intercept, y-intercept

3. Use slope properties for parallel and perpendicular lines

4. Practice two-point and point-slope forms

5. Solve previous year JEE coordinate geometry problems for speed

StudentBro.in JEE Maths Rectangular Cartesian Section

At StudentBro.in, students can access:

• Step-by-step explanations for distance, midpoint, slope, line, and triangle problems

• Worked examples for parallel, perpendicular lines, and intersections

• Practice questions with solutions and shortcuts

• Tips for quick problem-solving and exam strategy

• Revision notes for last-minute preparation

Practice Questions and Sample Problems

1. Find distance between points A(2, 5) and B(7, 9)

2. Find midpoint of line segment joining P(1, 2) and Q(5, 6)

3. Find slope of line passing through (3, 4) and (6, 8)

4. Find area of triangle with vertices A(0,0), B(4,0), C(0,3)

5. Distance of point (2,3) from line 3x + 4y - 5 = 0

Regular practice improves accuracy, speed, and confidence in solving coordinate geometry problems in JEE exams.

Conclusion

Rectangular Cartesian Coordinates are a fundamental chapter in JEE Maths. Understanding distance, midpoint, slope, equation of a line, and area formulas allows students to solve problems efficiently. At StudentBro.in, we provide a complete guide from basics to advanced problem-solving, making coordinate geometry simple, practical, and exam-oriented.

Mastering this topic ensures students can tackle lines, triangles, and points problems confidently in both JEE Main and Advanced exams.