Tangent & NormalJEE Main

Concept

The tangent at a point is the straight line that just grazes the curve there — its slope is the derivative $f'(x_0)$. The normal is perpendicular to it, with slope $-1/f'(x_0)$.

Key formula

Derivation

The derivative gives the instantaneous slope $m = f'(x_0)$. Using point–slope form through $(x_0, y_0)$ yields the tangent.

Perpendicular lines have slopes whose product is $-1$, so the normal's slope is $-1/m$. Where the tangent is horizontal ($m=0$), the normal is vertical.

Scenarios to explore

• Tangent & Normal — Lines touching a curve and the perpendicular to them.

Real-world applications

• Optics: normals define angles of incidence and reflection.
• Physics: velocity is tangent to a trajectory.
• Newton–Raphson root finding follows tangents to the x-axis.

JEE exam tips

• Tangent length, subtangent and subnormal all derive from $y_0$ and $f'(x_0)$.
• Horizontal tangent ⇒ $f'(x_0)=0$ (a turning point); vertical tangent ⇒ $f'$ infinite.

Common mistakes

• Using $f(x_0)$ as the slope instead of $f'(x_0)$.
• Forgetting the normal's slope is the negative reciprocal, not the negative.
• Point–slope sign errors in $(x - x_0)$.

Exam traps to avoid

• At a point where $f'(x_0)=0$, the normal is vertical (undefined slope), not horizontal.
• Two curves are orthogonal where their tangents' slopes multiply to $-1$.