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Straight Lines revision notes

Every idea in the Straight Lines chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 24 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Coordinates, section formula and area

Distance and section formula
10× asked, last 2025
AB=(x2−x1)2+(y2−y1)2,P=(mx2+nx1m+n, my2+ny1m+n)AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2},\qquad P=\left(\frac{mx_2+nx_1}{m+n},\ \frac{my_2+ny_1}{m+n}\right)
AP:PB=m:nAP:PB=m:n from A(x1,y1)A(x_1,y_1) to B(x2,y2)B(x_2,y_2). External division: replace nn by −n-n (m≠nm\ne n). Midpoint: m=nm=n.
xxyyA(1,1)A(1,1)B(7,4)B(7,4)P(3,2)P(3,2), 1:21:2QQ, 3:13:1 outside
AP:PB=1:2AP:PB=1:2 puts weight 2 on AA: P=2A+B3P=\frac{2A+B}3, nearer AA. External 3:13:1: Q=3B−A2Q=\frac{3B-A}{2}
  • Not (mA+nB)/(m+n)(mA+nB)/(m+n). For AP:PB=m:nAP:PB=m:n the weight mm goes on BB, the far end, so P=(nA+mB)/(m+n)P=(nA+mB)/(m+n). Check: PP must be nearer AA when m<nm<n.
Parallelogram and midpoints
8× asked, last 2026
A+C=B+D  (parallelogram ABCD),A=E+F−DA+C=B+D\ \ (\text{parallelogram } ABCD),\qquad A=E+F-D
Diagonals of any parallelogram bisect each other. D,E,FD,E,F = midpoints of BC,CA,ABBC,CA,AB; triangle DEFDEF has a quarter of the area.
xxyyA(1,2)A(1,2)C(−3,−6)C(-3,-6)BBDDM(−1,−2)M(-1,-2)
Diagonals share the midpoint MM: B+D=A+C=(−2,−4)B+D=A+C=(-2,-4)
AABBCCDDEEFF
Midpoint triangle DEFDEF: a quarter of the area
  • Not the midpoint's coordinate sum. B+D=A+CB+D=A+C is twice the midpoint. With A(1,2)A(1,2), C(−3,−6)C(-3,-6), the coordinates of BB and DD add up to −6-6, not −3-3.
Area of a triangle
13× asked, last 2025
Δ=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\Delta=\tfrac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|
Vertices in any order; the modulus keeps it positive. Three points are collinear exactly when this is 0.
xxyyE=+2ΔE=+2\DeltaE=−2ΔE=-2\Deltabase
Base fixed, area fixed: the third vertex lies on either of two lines parallel to the base
  • Not one case from ∣E∣=2Δ|E|=2\Delta. It means E=2ΔE=2\Delta or E=−2ΔE=-2\Delta: a fixed base and area put the third vertex on two lines parallel to the base. Keep both unless a side is named ('above ABAB').
  • Quick method. Triangles with a common vertex and bases on one line have areas in the ratio of the bases, so an area ratio is a section ratio. Sharing an angle: areas are in the product of the side ratios.
Rotation and reflection of a point
5× asked, last 2026
(x,y) → (xcos⁡θ−ysin⁡θ, xsin⁡θ+ycos⁡θ)(x,y)\ \to\ (x\cos\theta-y\sin\theta,\ x\sin\theta+y\cos\theta)
Anticlockwise by θ\theta about the origin. Images: in y=xy=x (y,x)(y,x); in the xx-axis (x,−y)(x,-y); in the origin (−x,−y)(-x,-y).
xxyyy=xy=x90∘90^\circ(3,1)(3,1)(−1,3)(-1,3): turned 90∘90^\circ(1,3)(1,3) in y=xy=x(3,−1)(3,-1)(−3,−1)(-3,-1) in O
Turning about O keeps the distance from O; the open dots are the images in y=xy=x, the xx-axis and O
  • Quick method. Equilateral PQRPQR with QQ, RR on two parallel lines: put Q−P=a(cos⁡θ,sin⁡θ)Q-P=a(\cos\theta,\sin\theta) and turn it by 60∘60^\circ for R−PR-P. The two distances give asin⁡θa\sin\theta and acos⁡θa\cos\theta; add their squares.

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Practise Straight Lines

20 more concepts in 6 topics

Centres of a triangle; Slope, angle and forms of a line; Intersection, concurrency and position; Distances, foot and image; Angle bisectors and family of lines; and Locus. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Centres of a triangle

  • Centroid10×
  • Incentre and inradius3×
  • Circumcentre7×
  • Orthocentre9×
  • Euler line6×

Slope, angle and forms of a line

  • Slope and angle between two lines11×
  • Slope, point and two-point forms6×
  • Intercept form7×
  • Normal form4×
  • Parametric (distance) form4×

Intersection, concurrency and position

  • Intersection of two lines8×
  • Concurrency of three lines3×
  • Position of a point relative to a line6×

Distances, foot and image

  • Distance of a point from a line17×
  • Distance between parallel lines5×
  • Foot of perpendicular and image11×
  • Reflection of a light ray7×

Angle bisectors and family of lines

  • Angle bisectors of two lines10×
  • Family of lines through an intersection4×

Locus

  • Finding a locus13×

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