Coordinate Geometry – JEE Mains Mathematics

1. Cartesian Coordinate System

The coordinate plane consists of two perpendicular axes: the **x-axis** (horizontal) and the **y-axis** (vertical).
Each point in the plane is represented by an ordered pair (x, y), where x is the coordinate along the x-axis, and y is the coordinate along the y-axis.

2. Distance Formula

The distance between two points (x₁, y₁) and (x₂, y₂) is given by:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

3. Section Formula

The section formula is used to find the coordinates of a point that divides the line segment joining two points in a given ratio.
If a point P divides the line segment joining A(x₁, y₁) and B(x₂, y₂) in the ratio m:n, then the coordinates of P are:

P = ( (mx₂ + nx₁) / (m + n), (my₂ + ny₁) / (m + n) )

4. Locus and its Equation

The locus of a point is the set of all points that satisfy a given condition.
The equation of the locus represents the geometric place of points that satisfy the condition.
For example, the equation of a circle is the locus of points equidistant from a fixed point (the center).

5. Slope of a Line

The slope (m) of a line is defined as the tangent of the angle θ that the line makes with the positive direction of the x-axis.

Slope, m = (y₂ - y₁) / (x₂ - x₁)

If two lines are parallel, their slopes are equal. If two lines are perpendicular, the product of their slopes is -1.

6. Parallel and Perpendicular Lines

Two lines are parallel if they have the same slope.
Two lines are perpendicular if the product of their slopes is -1.

7. Intercepts of a Line on the Coordinate Axis

The x-intercept of a line is the point where the line intersects the x-axis, and similarly, the y-intercept is where the line intersects the y-axis.
If the equation of a line is in the form Ax + By + C = 0, the intercepts are:

x-intercept = -C / A, y-intercept = -C / B

8. Equation of a Straight Line

The equation of a straight line can be represented in various forms:

**Point-Slope Form**: y - y₁ = m(x - x₁)
**Slope-Intercept Form**: y = mx + c
**General Form**: Ax + By + C = 0

9. Angle Between Two Lines

The angle θ between two lines with slopes m₁ and m₂ is given by:

tan(θ) = |(m₁ - m₂) / (1 + m₁m₂)|

10. Concurrence of Three Lines

Three lines are said to be concurrent if they intersect at a single point.
For three lines with equations L₁: A₁x + B₁y + C₁ = 0, L₂: A₂x + B₂y + C₂ = 0, and L₃: A₃x + B₃y + C₃ = 0, they are concurrent if the determinant of the coefficients is zero:

Determinant = |A₁ B₁ C₁| = 0
|A₂ B₂ C₂|
|A₃ B₃ C₃|

11. Distance of a Point from a Line

The distance of a point P(x₁, y₁) from a line Ax + By + C = 0 is given by:

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

12. Centroid, Orthocenter, and Circumcenter of a Triangle

The **centroid** of a triangle is the point of intersection of the medians, and it divides each median in a 2:1 ratio.
The **orthocenter** is the point of intersection of the altitudes of a triangle.
The **circumcenter** is the point of intersection of the perpendicular bisectors of the sides of the triangle, and it is equidistant from all three vertices of the triangle.

13. Circle

The equation of a circle with center (h, k) and radius r is:

(x - h)² + (y - k)² = r²

The general form of the equation of a circle is:

x² + y² + 2gx + 2fy + c = 0

The center is (-g, -f) and the radius is √(g² + f² - c).
If the endpoints of a diameter of a circle are given, the equation of the circle can be derived using the midpoint formula for the center and the distance formula for the radius.

14. Conic Sections

Conic sections are the curves formed by the intersection of a plane and a cone. The three primary types of conic sections are:

**Parabola**: The equation of a parabola in standard form is y² = 4ax or x² = 4ay.
**Ellipse**: The standard form of an ellipse is (x² / a²) + (y² / b²) = 1.
**Hyperbola**: The standard form of a hyperbola is (x² / a²) - (y² / b²) = 1.