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JEE Mains Maths · Formula sheet

Straight Lines formulas

17 formulas and 17 common traps for JEE Mains Maths Straight Lines, grouped by subtopic.

Full notes with worked examples

Slope, Angle and Forms of a Line

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The angle between two lines

Angle between two lines

tan⁡θ=∣m1−m21+m1m2∣\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|

Intercept and normal forms

xa+yb=1,xcos⁡α+ysin⁡α=p\frac xa+\frac yb=1,\qquad x\cos\alpha+y\sin\alpha=p

Distance along a line

Parametric form

x−x1cos⁡θ=y−y1sin⁡θ=r\frac{x-x_1}{\cos\theta}=\frac{y-y_1}{\sin\theta}=r

Common traps

Two lines, not one

tan⁡θ=k\tan\theta=k with a modulus gives two slopes. An isosceles triangle with two sides given has two possible bases, and a line at a given angle to another has two positions. Keep both before you add or choose.

Normal form needs a unit normal

3x+4y=103x+4y=10 is not in normal form until you divide by 32+42=5\sqrt{3^2+4^2}=5: 35x+45y=2\frac35x+\frac45y=2, so p=2p=2. Reading p=10p=10 from the raw equation is the usual slip.

Two directions give two answers

When only the distance is given, θ\theta is the unknown, and the equation for it usually has two solutions: two lines from the point reach the line at that distance. The question may want both, or their sum.

Distance from a Line and Parallel Lines

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Distance from a point to a line

Perpendicular distance

d=∣ax1+by1+c∣a2+b2d=\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}

Parallel lines

Distance between parallel lines

d=∣c1−c2∣a2+b2d=\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}

Which side of a line

Same side of a line

(ax1+by1+c)(ax2+by2+c)>0(ax_1+by_1+c)(ax_2+by_2+c)>0

Common traps

Write the line as ax + by + c = 0 first

The denominator is a2+b2\sqrt{a^2+b^2} of the rearranged equation. For y=2x+3y=2x+3, that is 2x−y+3=02x-y+3=0, so the denominator is 5\sqrt5, not 33.

Match the coefficients first

x+2y+1=0x+2y+1=0 and 2x+4y+7=02x+4y+7=0 are parallel, but the formula needs equal aa and bb. Doubling the first gives ∣2−7∣20\frac{|2-7|}{\sqrt{20}}, not ∣1−7∣5\frac{|1-7|}{\sqrt5}.

Test with the vertex, not the sketch

Deciding "inside" from a rough picture fails when the lines are close together. For each side, compare signs with the opposite vertex; the three inequalities give the exact range.

Image of a Point and Reflected Rays

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Image of a point in a line

Image in a line

x−x1a=y−y1b=−2(ax1+by1+c)a2+b2\frac{x-x_1}{a}=\frac{y-y_1}{b}=-\frac{2(ax_1+by_1+c)}{a^2+b^2}

Reflected rays and shortest paths

Path via a mirror

AR+RB≥A′B, equality when R is on A′BAR+RB\ge A'B,\ \text{equality when } R \text{ is on } A'B

Common traps

Twice, not once

The formula with 1 gives the foot of the perpendicular; with 2 it gives the image. Check the answer: the midpoint of PP and its image must lie on the line.

Reflect the right point

The reflected ray passes through the image of the source. Reflecting the far point instead gives the incident ray. Check which ray the question asks for before you reflect.

Angle Bisectors and Pairs of Lines

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Angle bisectors

a1x+b1y+c1a12+b12=±a2x+b2y+c2a22+b22\frac{a_1x+b_1y+c_1}{\sqrt{a_1^2+b_1^2}}=\pm\frac{a_2x+b_2y+c_2}{\sqrt{a_2^2+b_2^2}}

A pair of lines in one equation

Pair of lines through the origin

ax2+2hxy+by2=0:tan⁡θ=2h2−ab∣a+b∣ax^2+2hxy+by^2=0:\quad \tan\theta=\frac{2\sqrt{h^2-ab}}{|a+b|}

Common traps

Origin angle is not always the acute one

The + sign picks the angle containing the origin, which may be acute or obtuse. For the acute bisector, check the angle it makes with one of the lines: it must be less than 45∘45^\circ.

h is half the xy coefficient

In ax2+2hxy+by2ax^2+2hxy+by^2, the coefficient of xyxy is 2h2h. For x2−4xy+y2x^2-4xy+y^2, h=−2h=-2, not −4-4.

Centres of a Triangle

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The orthocentre

Altitude conditions

AH→⋅BC→=0,BH→⋅CA→=0\overrightarrow{AH}\cdot\overrightarrow{BC}=0,\qquad \overrightarrow{BH}\cdot\overrightarrow{CA}=0

The circumcentre

Equal distances

∣OA∣2=∣OB∣2=∣OC∣2|OA|^2=|OB|^2=|OC|^2

Centroid, incentre and dividing a side

Incentre

I=aA+bB+cCa+b+cI=\frac{aA+bB+cC}{a+b+c}

Common traps

Check for a right angle first

If two sides have slopes whose product is −1-1, the orthocentre is their common vertex, and no altitude needs writing. Missing this turns a one-line question into a page of algebra.

Midpoint alone is not enough

The perpendicular bisector needs the midpoint and the perpendicular direction. A median also passes through the midpoint, but it goes through OO only when the triangle is isosceles there.

Weight by the opposite side

In the incentre formula, AA is weighted by a=BCa=BC, the side opposite it, not by a side through it. Weighting by the adjacent sides gives a different point.

Area of Triangles and Quadrilaterals

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Area from coordinates

Area of a triangle

Δ=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\Delta=\frac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|

Parallelograms, rhombi and isosceles triangles

Parallelogram

A+C=B+DA+C=B+D

Common traps

Two signs from one modulus

An area of 4 means the bracket is 88 or −8-8. Dropping one sign loses half the answers, which matters when the question asks for the sum of all values.

Which vertices are opposite

A+C=B+DA+C=B+D holds when AA and CC are opposite. If the vertices are named ABCDABCD in order, they are. If only three points are given, the fourth vertex has three possible positions.

Family of Lines and Locus

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Lines through a fixed point

Family through a point

L1+λL2=0L_1+\lambda L_2=0

Locus of a moving point

Elimination

h=f(t), k=g(t) ⇒ F(h,k)=0h=f(t),\ k=g(t)\ \Rightarrow\ F(h,k)=0

Common traps

Count every way to fail

Three lines do not form a triangle when any two are parallel or when all three meet at one point. Each case gives its own values of the parameter, and the answer needs all of them.

Keep the restrictions

Eliminating the parameter can add points the moving point never reaches. If θ\theta or a coordinate is restricted, the locus is only the matching part of the curve, and the options may differ only there.

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