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Tangent AND Normal formulas

Master Tangent AND Normal through 34 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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Tangent AND Normal, every formula

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Slope of tangent
$m = tan θ = \frac{d y}{d x}_{(x_{1}, y_{1})}$
Slope for f(x,y)=0
$m = - (\frac{\partial f / \partial x}{\partial f / \partial y})_{(x_{1}, y_{1})}$
Q1MCQImplicit slope
For $f(x,y)=0$, the tangent slope is:
- A
  $-\dfrac{f_x}{f_y}$
- B
  $\dfrac{f_x}{f_y}$
- C
  $-\dfrac{f_y}{f_x}$
- D
  $f_x f_y$
Slope (parametric)
$m = \frac{g ^{'} ( t )}{f ^{'} ( t )}$
x=f(t), y=g(t)
Q1MCQParametric slope
For $x=t^{2},\ y=t^{3}$, the slope $\dfrac{dy}{dx}$ is:
- A
  $\dfrac{3t}{2}$
- B
  $\dfrac{2}{3t}$
- C
  $3t^{2}$
- D
  $\dfrac{2t}{3}$
Tangent parallel to x-axis
$\frac{d y}{d x} = 0$
Tangent perpendicular to x-axis
$\frac{d y}{d x} \to \infty or \frac{d x}{d y} = 0$
Q1MCQPerpendicular to x-axis
A tangent is perpendicular to the x-axis when:
- A
  $\dfrac{dy}{dx}\to\infty$
- B
  $\dfrac{dy}{dx}=0$
- C
  $\dfrac{dy}{dx}=1$
- D
  $\dfrac{dy}{dx}=-1$
Equation of tangent
$y - y_{1} = m (x - x_{1})$
Q1MCQTangent equation
The tangent to $y=x^{2}$ at $(1,1)$ (slope $2$) is:
- A
  $y-1=2(x-1)$
- B
  $y-1=-\dfrac12(x-1)$
- C
  $y=x$
- D
  $y-1=x-1$
Equation of normal
$y - y_{1} = - \frac{1}{m} (x - x_{1})$
Q1MCQNormal equation
The normal to $y=x^{2}$ at $(1,1)$ (slope of tangent $2$) is:
- A
  $y-1=-\dfrac12(x-1)$
- B
  $y-1=2(x-1)$
- C
  $y-1=\dfrac12(x-1)$
- D
  $y=2x$
Slope of normal
$- \frac{1}{d y / d x}$
Q1MCQSlope of normal
If the tangent slope is $m$, the normal slope is:
- A
  $-\dfrac1m$
- B
  $m$
- C
  $\dfrac1m$
- D
  $-m$
Tangent at origin (lowest degree)
$equate lowest-degree terms to 0$
Q1MCQTangent at origin
For $x^{2}+y^{2}+2gx+2fy=0$, the tangent at the origin is:
- A
  $gx+fy=0$
- B
  $x^{2}+y^{2}=0$
- C
  $gx-fy=0$
- D
  $x+y=0$
Equally inclined tangent
$\frac{d y}{d x} = \pm 1$
Q1MCQEqually inclined
A tangent equally inclined to the axes has slope:
- A
  $\pm1$
- B
  $0$
- C
  $\infty$
- D
  $2$
Length of tangent
$P T = \frac{y _{1} 1 + m ^{2}}{m}$
Q1MCQLength of tangent
The length of the tangent $PT$ is:
- A
  $\left|\dfrac{y_1\sqrt{1+m^{2}}}{m}\right|$
- B
  $|y_1\sqrt{1+m^{2}}|$
- C
  $\left|\dfrac{y_1}{m}\right|$
- D
  $|y_1 m|$
Length of normal
$P N = y_{1} 1 + m^{2}$
Q1MCQLength of normal
The length of the normal $PN$ is:
- A
  $|y_1\sqrt{1+m^{2}}|$
- B
  $\left|\dfrac{y_1\sqrt{1+m^{2}}}{m}\right|$
- C
  $|y_1 m|$
- D
  $\left|\dfrac{y_1}{m}\right|$
Length of sub-tangent
$T M = \frac{y _{1}}{m}$
Q1MCQSub-tangent
The length of the sub-tangent is:
- A
  $\left|\dfrac{y_1}{m}\right|$
- B
  $|y_1 m|$
- C
  $|y_1|$
- D
  $|m|$
Length of sub-normal
$M N = ∣ y_{1} m ∣$
Q1MCQSub-normal
The length of the sub-normal is:
- A
  $|y_1 m|$
- B
  $\left|\dfrac{y_1}{m}\right|$
- C
  $|y_1|$
- D
  $|m|$
Sub-tangent/normal in GP
$y_{1}^{2} = (sub-tangent) (sub-normal)$
Q1MCQGP relation
For $y=f(x)$, (ordinate)$^{2}$ equals:
- A
  (sub-tangent)(sub-normal)
- B
  (sub-tangent)$+$(sub-normal)
- C
  (tangent)(normal)
- D
  $m^{2}$
Angle between two curves
$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$
Q1MCQAngle between curves
The angle between two curves with tangent slopes $m_1,m_2$ satisfies:
- A
  $\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1m_2}\right|$
- B
  $\tan\theta=|m_1m_2|$
- C
  $\tan\theta=|m_1+m_2|$
- D
  $\cos\theta=m_1m_2$
Curves touch
$m_{1} = m_{2}$
Q1MCQCurves touch
Two curves touch each other at $P$ if:
- A
  $m_1=m_2$
- B
  $m_1m_2=-1$
- C
  $m_1+m_2=0$
- D
  $m_1=0$
Orthogonal curves
$m_{1} m_{2} = - 1$
Q1MCQOrthogonal
Two curves cut orthogonally if:
- A
  $m_1m_2=-1$
- B
  $m_1=m_2$
- C
  $m_1+m_2=1$
- D
  $m_1m_2=1$
Orthogonal (implicit)
$\frac{\partial f}{\partial x} \frac{\partial g}{\partial x} + \frac{\partial f}{\partial y} \frac{\partial g}{\partial y} = 0$
Q1MCQOrthogonal implicit
Curves $f=0,\ g=0$ cut orthogonally if:
- A
  $f_x g_x+f_y g_y=0$
- B
  $f_x g_x-f_y g_y=0$
- C
  $f_x g_y=f_y g_x$
- D
  $f_x+g_x=0$
Rate of change
$\frac{d y}{d t} = \frac{d y}{d x} \cdot \frac{d x}{d t}$
Q1MCQRate of change
$\dfrac{dy}{dt}$ equals:
- A
  $\dfrac{dy}{dx}\cdot\dfrac{dx}{dt}$
- B
  $\dfrac{dy}{dx}$
- C
  $\dfrac{dx}{dt}$
- D
  $\dfrac{dt}{dx}$
Velocity
$v = \frac{d s}{d t}$
Acceleration
$a = \frac{d v}{d t} = \frac{d ^{2} s}{d t ^{2}} = v \frac{d v}{d s}$
Differential (approximation)
$d y = f^{'} (x) δ x$
Approx value
$f (x + δ x) \approx f (x) + f^{'} (x) δ x$
Q1MCQApprox value
The approximation $f(x+\delta x)\approx$:
- A
  $f(x)+f'(x)\delta x$
- B
  $f(x)$
- C
  $f'(x)\delta x$
- D
  $f(x)\delta x$
Relative error
$\frac{δ y}{y}$
Q1MCQRelative error
The relative error in $y$ is:
- A
  $\dfrac{\delta y}{y}$
- B
  $\delta y$
- C
  $\dfrac{\delta y}{y}\times100$
- D
  $y\,\delta y$
Percentage error
$\frac{δ y}{y} \times 100$
Q1MCQPercentage error
The percentage error in $y$ is:
- A
  $\dfrac{\delta y}{y}\times100$
- B
  $\dfrac{\delta y}{y}$
- C
  $\delta y$
- D
  $100\,\delta y$
Error in kxⁿ
$\frac{δ y}{y} = n \frac{δ x}{x}$
f=kxⁿ
Area of triangle (xy=c²)
$2 c^{2} sq. units$
tangent + axes
Q1MCQTriangle area xy=c²
The area of the triangle formed by a tangent to $xy=c^{2}$ and the axes is:
- A
  $2c^{2}$
- B
  $c^{2}$
- C
  $4c^{2}$
- D
  $\dfrac{c^{2}}{2}$
Sphere surface & volume
$S = 4 π r^{2}, V = \frac{4}{3} π r^{3}$
Q1MCQSphere
The volume of a sphere of radius $r$ is:
- A
  $\dfrac43\pi r^{3}$
- B
  $4\pi r^{2}$
- C
  $\pi r^{2}h$
- D
  $\dfrac13\pi r^{2}h$
Cylinder surface & volume
$S = 2 π r h + 2 π r^{2}, V = π r^{2} h$
Q1MCQCylinder
The volume of a right circular cylinder is:
- A
  $\pi r^{2}h$
- B
  $2\pi rh$
- C
  $\dfrac13\pi r^{2}h$
- D
  $\dfrac43\pi r^{3}$
Cone: l, curved & total surface
$l^{2} = r^{2} + h^{2}, CSA = π r l, TSA = π r^{2} + π r l$
Q1MCQCone slant height
For a cone, $l^{2}$ equals:
- A
  $r^{2}+h^{2}$
- B
  $r^{2}-h^{2}$
- C
  $2rh$
- D
  $r^{2}h^{2}$
Cone volume
$V = \frac{1}{3} π r^{2} h$
Q1MCQCone volume
The volume of a cone is:
- A
  $\dfrac13\pi r^{2}h$
- B
  $\pi r^{2}h$
- C
  $\dfrac43\pi r^{3}$
- D
  $\pi rl$
Sector arc, area
$l = r θ, A = \frac{1}{2} r^{2} θ$
Q1MCQSector area
The area of a sector with radius $r$ and angle $\theta$ is:
- A
  $\dfrac12 r^{2}\theta$
- B
  $r\theta$
- C
  $\pi r^{2}$
- D
  $\dfrac12 r\theta$
Simple pendulum period
$T = 2 π \frac{l}{g}$
Q1MCQPendulum period
The period of a simple pendulum of length $l$ is:
- A
  $2\pi\sqrt{\dfrac{l}{g}}$
- B
  $2\pi\sqrt{\dfrac{g}{l}}$
- C
  $\pi\sqrt{\dfrac{l}{g}}$
- D
  $2\pi\dfrac{l}{g}$

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Tangent AND Normal formulas

Tangent AND Normal, every formula

Slope of tangent

Slope for f(x,y)=0

Slope (parametric)

Tangent parallel to x-axis

Tangent perpendicular to x-axis

Equation of tangent

Equation of normal

Slope of normal

Tangent at origin (lowest degree)

Equally inclined tangent

Length of tangent

Length of normal

Length of sub-tangent

Length of sub-normal

Sub-tangent/normal in GP

Angle between two curves

Curves touch

Orthogonal curves

Orthogonal (implicit)

Rate of change

Velocity

Acceleration

Differential (approximation)

Approx value

Relative error

Percentage error

Error in kxⁿ

Area of triangle (xy=c²)

Sphere surface & volume

Cylinder surface & volume

Cone: l, curved & total surface

Cone volume

Sector arc, area

Simple pendulum period

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