Straight Lines - How It Appears in JEE

6 recurring patterns. Learn the pattern, recognize it in 5 seconds, apply the right approach.

Pattern

Find distance from point to line, between parallel lines, or position of points relative to lines

How to recognize

Asks for distance, foot of perpendicular, or which side of a line a point lies on

→ Use d = |ax₁+by₁+c|/√(a²+b²). For position, check the sign of ax₁+by₁+c for the point vs origin.

Pattern

Find angle between lines, or equation after rotation about a point

How to recognize

Mentions angle between lines, rotation, tan of angle, perpendicular/parallel conditions

→ Use tanθ = |m₁-m₂|/(1+m₁m₂). For rotation by α: new slope = tan(θ±α) where θ is original angle.

Pattern

Find centroid, circumcenter, orthocenter, incenter, or area of a triangle

How to recognize

Given vertices, asks for special points, area, or properties of the triangle

→ Centroid = average of vertices. Orthocenter: intersection of altitudes. Circumcenter: equidistant from vertices.

Pattern

Find the equation of locus satisfying a geometric condition

How to recognize

Point moves according to a condition, find its path/locus equation

→ Let point be (h,k). Apply the condition. Replace h→x, k→y to get the locus equation.

Pattern

Check concurrency or find a line through intersection of two given lines

How to recognize

Three or more lines, asks if concurrent, or line through intersection satisfying a condition

→ Concurrency: determinant of coefficients = 0. Family: L₁ + λL₂ = 0, use extra condition to find λ.

Pattern

Find image of a point or line in another line, or reflection problems

How to recognize

Image, reflection, mirror line, incident/reflected ray

→ Image formula: (x-x₁)/a = (y-y₁)/b = -2(ax₁+by₁+c)/(a²+b²). For reflected ray: reflect point across mirror line.