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Straight Line formulas

Master Straight Line through 32 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Straight Line, every formula

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Distance between two points
$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Section formula (internal)
$(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$
Q1MCQSection internal
The point dividing $(1,2)$ and $(4,8)$ in ratio $1:2$ internally is:
- A
  $(2,4)$
- B
  $(3,6)$
- C
  $(2.5,5)$
- D
  $(1.5,3)$
Section formula (external)
$(\frac{m x _{2} - n x _{1}}{m - n}, \frac{m y _{2} - n y _{1}}{m - n})$
Q1MCQSection external
External division uses:
- A
  $\dfrac{mx_2-nx_1}{m-n}$
- B
  $\dfrac{mx_2+nx_1}{m+n}$
- C
  $\dfrac{x_1+x_2}{2}$
- D
  $mx_2-nx_1$
Midpoint
$(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
Q1MCQMidpoint
The midpoint of $(2,6)$ and $(4,2)$ is:
- A
  $(3,4)$
- B
  $(6,8)$
- C
  $(1,2)$
- D
  $(2,4)$
Centroid of a triangle
$(\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3})$
Q1MCQCentroid
The centroid of $(0,0),(3,0),(0,3)$ is:
- A
  $(1,1)$
- B
  $(1.5,1.5)$
- C
  $(3,3)$
- D
  $(0,0)$
Area of a triangle
$\frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣$
Collinearity
$area = 0$
three points
Q1MCQCollinearity
Three points are collinear iff the area of the triangle they form is:
- A
  $0$
- B
  $1$
- C
  $\infty$
- D
  positive
Slope
$m = tan θ = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Slope-intercept form
$y = m x + c$
Q1MCQSlope-intercept
The slope of $y=2x+5$ is:
- A
  $2$
- B
  $5$
- C
  $-2$
- D
  $\dfrac12$
Point-slope form
$y - y_{1} = m (x - x_{1})$
Q1MCQPoint-slope
The line through $(1,3)$ with slope $2$ is:
- A
  $y-3=2(x-1)$
- B
  $y-1=2(x-3)$
- C
  $y=2x$
- D
  $y+3=2(x+1)$
Two-point form
$y - y_{1} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} (x - x_{1})$
Q1MCQTwo-point
The line through $(0,0)$ and $(2,4)$ is:
- A
  $y=2x$
- B
  $y=x$
- C
  $y=4x$
- D
  $x=2y$
Intercept form
$\frac{x}{a} + \frac{y}{b} = 1$
Q1MCQIntercept form
The line $\dfrac{x}{2}+\dfrac{y}{3}=1$ has x-intercept:
- A
  $2$
- B
  $3$
- C
  $1$
- D
  $6$
Normal form
$x cos α + y sin α = p$
p = distance from origin
Q1MCQNormal form
In $x\cos\alpha+y\sin\alpha=p$, $p$ represents:
- A
  the perpendicular distance from the origin
- B
  the slope
- C
  the x-intercept
- D
  the y-intercept
Slope from general form
$a x + b y + c = 0 \Rightarrow m = - \frac{a}{b}$
Q1MCQSlope from general form
The slope of $3x+4y-12=0$ is:
- A
  $-\dfrac34$
- B
  $\dfrac34$
- C
  $-\dfrac43$
- D
  $\dfrac43$
Angle between two lines
$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$
Parallel lines
$m_{1} = m_{2}$
Q1MCQParallel
A line parallel to $y=3x+1$ has slope:
- A
  $3$
- B
  $-\dfrac13$
- C
  $1$
- D
  $-3$
Perpendicular lines
$m_{1} m_{2} = - 1$
Q1MCQPerpendicular
A line perpendicular to $y=2x$ has slope:
- A
  $-\dfrac12$
- B
  $2$
- C
  $\dfrac12$
- D
  $-2$
Distance point to line
$\frac{∣ a x _{1} + b y _{1} + c ∣}{a ^{2} + b ^{2}}$
Distance between parallel lines
$\frac{∣ c _{1} - c _{2} ∣}{a ^{2} + b ^{2}}$
Perpendicular distance from origin
$\frac{∣ c ∣}{a ^{2} + b ^{2}}$
Line parallel to ax+by+c=0
$a x + b y + k = 0$
Q1MCQParallel line eqn
A line parallel to $2x+3y+1=0$ has the form:
- A
  $2x+3y+k=0$
- B
  $3x-2y+k=0$
- C
  $2x-3y+k=0$
- D
  $3x+2y+k=0$
Line perpendicular to ax+by+c=0
$b x - a y + k = 0$
Q1MCQPerpendicular line eqn
A line perpendicular to $2x+3y+1=0$ has the form:
- A
  $3x-2y+k=0$
- B
  $2x+3y+k=0$
- C
  $2x-3y+k=0$
- D
  $3x+2y+k=0$
Family of lines through intersection
$L_{1} + λ L_{2} = 0$
Q1MCQFamily of lines
A line through the intersection of $L_1=0$ and $L_2=0$ is:
- A
  $L_1+\lambda L_2=0$
- B
  $L_1 L_2=0$
- C
  $L_1-L_2=1$
- D
  $L_1=L_2$
Concurrency of three lines
$a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} = 0$
Q1MCQConcurrency
Three lines are concurrent iff the determinant of their coefficients is:
- A
  $0$
- B
  $1$
- C
  $\neq0$
- D
  $\infty$
Position of two points w.r.t. a line
$(a x_{1} + b y_{1} + c) (a x_{2} + b y_{2} + c) > 0 same side$
Q1MCQSame/opposite side
Two points lie on the same side of $ax+by+c=0$ iff $(ax_1+by_1+c)(ax_2+by_2+c)$ is:
- A
  $>0$
- B
  $<0$
- C
  $=0$
- D
  $=1$
Angle bisectors
$\frac{a _{1} x + b _{1} y + c _{1}}{a _{1}^{2} + b _{1}^{2}} = \pm \frac{a _{2} x + b _{2} y + c _{2}}{a _{2}^{2} + b _{2}^{2}}$
Q1MCQAngle bisectors
The angle bisectors of two lines are given by:
- A
  $\dfrac{a_1x+b_1y+c_1}{\sqrt{a_1^{2}+b_1^{2}}}=\pm\dfrac{a_2x+b_2y+c_2}{\sqrt{a_2^{2}+b_2^{2}}}$
- B
  $L_1=L_2$
- C
  $L_1+L_2=0$
- D
  $L_1 L_2=0$
Foot of perpendicular
$\frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = - \frac{a x _{1} + b y _{1} + c}{a ^{2} + b ^{2}}$
Q1MCQFoot of perpendicular
The foot of the perpendicular from a point uses the parameter:
- A
  $-\dfrac{ax_1+by_1+c}{a^{2}+b^{2}}$
- B
  $-\dfrac{2(ax_1+by_1+c)}{a^{2}+b^{2}}$
- C
  $\dfrac{c}{\sqrt{a^{2}+b^{2}}}$
- D
  $0$
Image of a point in a line
$\frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = - \frac{2 ( a x _{1} + b y _{1} + c )}{a ^{2} + b ^{2}}$
Q1MCQImage of a point
The image of a point in a line uses the parameter:
- A
  $-\dfrac{2(ax_1+by_1+c)}{a^{2}+b^{2}}$
- B
  $-\dfrac{ax_1+by_1+c}{a^{2}+b^{2}}$
- C
  $\dfrac{c}{a^{2}+b^{2}}$
- D
  $1$
Distance (parametric) form
$\frac{x - x _{1}}{cos θ} = \frac{y - y _{1}}{sin θ} = r$
Q1MCQDistance form
A point at distance $r$ along a line at angle $\theta$ from $(x_1,y_1)$ is:
- A
  $(x_1+r\cos\theta,\ y_1+r\sin\theta)$
- B
  $(x_1+r,\ y_1+r)$
- C
  $(r\cos\theta,\ r\sin\theta)$
- D
  $(x_1\cos\theta,\ y_1\sin\theta)$
Slope of line through origin at angle θ
$y = x tan θ$
Q1MCQLine through origin
A line through the origin making $45^{\circ}$ with the x-axis is:
- A
  $y=x$
- B
  $y=-x$
- C
  $x=0$
- D
  $y=2x$
Line making equal intercepts
$x + y = a$
Q1MCQEqual intercepts
A line with equal non-zero intercepts through $(1,3)$ is:
- A
  $x+y=4$
- B
  $x-y=4$
- C
  $x+y=1$
- D
  $y=3x$
Reflection of x-axis line slope
$slope of tangent \to - slope for reflection in x-axis$
Q1MCQReflection in x-axis
Reflecting a line of slope $m$ in the x-axis gives slope:
- A
  $-m$
- B
  $m$
- C
  $\dfrac1m$
- D
  $-\dfrac1m$

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Straight Line formulas

Straight Line, every formula

Distance between two points

Section formula (internal)

Section formula (external)

Midpoint

Centroid of a triangle

Area of a triangle

Collinearity

Slope

Slope-intercept form

Point-slope form

Two-point form

Intercept form

Normal form

Slope from general form

Angle between two lines

Parallel lines

Perpendicular lines

Distance point to line

Distance between parallel lines

Perpendicular distance from origin

Line parallel to ax+by+c=0

Line perpendicular to ax+by+c=0

Family of lines through intersection

Concurrency of three lines

Position of two points w.r.t. a line

Angle bisectors

Foot of perpendicular

Image of a point in a line

Distance (parametric) form

Slope of line through origin at angle θ

Line making equal intercepts

Reflection of x-axis line slope

More JEE Advanced Mathematics formula sheets

Other ways to revise this chapter

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