Tangent and Normal Formulas

Take the help of Tangent and Normal Formulae to solve problems right from basic to an advanced level easily. Master the concept of Tangents and Normals with the provided formulae. Simplify the problems easily by applying the Tangents and Normal Formulas and cut through the hassle of doing lengthy calculations.

1. Geometrical interpretation of the derivative
If y = f(x) be a given function, then the differential coefficient f'(x) or \(\frac{d y}{d x}\) at the point P (x₁, y₁) is the trigonometrical tangent of the angle ψ (say) which the positive direction of the tangent to the curve at P makes with the positive direction of x-axis \(\left(\frac{d y}{d x}\right)\), therefore represents the slope of the tangent.
f'(x) = \(\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}\) = tan ψ
(i) The inclination of tangent with x-axis = tan^-1\(\left(\frac{d y}{d x}\right)\)
(ii) Slope of tangent = \(\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}\)
(iii) Slope of the normal = – \(\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}\)

2. Equation of tangent
Equation of tangent to the curve y = f(x) at P (x₁, y₁) is
y – y₁ = \(\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}\)(x – x₁)

• The tangent at (x₁, y₁) is parallel to x-axis ⇒ \(\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}\) = 0
• The tangent at (x₁, y₁) is parallel to y-axis ⇒ \(\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}\) = ∞
• The tangent line makes equal angles with the axes ⇒ \(\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}\) = ±1

3. Length of intercepts made on axes by the tangent

x-intercept = OA = x₁ – \(\left\{\frac{y_{1}}{\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}}\right\}\)
y-intercept = OB = y₁ – x₁\(\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}\)

4. Length of perpendicular from origin to the tangent
The length of perpendicular from origin (0, 0) to the tangent drawn at the point (x₁, y₁) of the curve y = f(x) is
p = \(\left|\frac{y_{1}-x_{1}\left(\frac{d y}{d x}\right)}{\sqrt{1+\left(\frac{d y}{d x}\right)^{2}}}\right|\)

5. Equation of Normal
The equation of normal at (x₁, y₁) to the curve y = f(x) is
(y – y₁) = – \(\frac{1}{\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}}\)(x – x₁)

6. Some facts about the normal
(i) The slope of the normal drawn at point P (x₁, y₁) to the curve y = f(x) is –\(\left(\frac{d x}{d y}\right)_{\left(x_{1}, y_{1}\right)}\)
(ii) If normal makes an angle of 0 with positive direction of x- axis then ⇒ \(\frac{d y}{d x}\) = – cot θ
(iii) If normal is parallel to x-axis then ⇒ \(\frac{d y}{d x}\) = ∞
(iv) If normal is parallel to y-axis then ⇒ \(\frac{d y}{d x}\) = 0
(v) If normal is equally inclined from both the axes or cuts equal intercept then ⇒ \(\frac{d y}{d x}\) = ± 1
(vi) The length of perpendicular from origin to normal is
p’ = \(\left|\frac{x_{1}+y_{1}\left(\frac{d y}{d x}\right)}{\sqrt{1+\left(\frac{d y}{d x}\right)^{2}}}\right|\)
(vii) The length of intercept made by normal on x-axis is x₁ + y₁\(\frac{d y}{d x}\) and length of intercept on y-axis is y₁ + x₁\(\frac{d y}{d x}\)

7. Equation of tangent and normal in “Parametric form”
If x = f(t) and y = g(t) then equation of tangent is a'(t)
(y – g(t)) = \(\frac{g^{\prime}(t)}{f^{\prime}(t)}\)(x – f(t)) and equation of normal is
(y – g(t)) = \(-\frac{f^{\prime}(t)}{g^{\prime}(t)}\)(x – f(t))

8. Angle of intersection of two curves
If two curves y = f₁(x) and y = f₂(x) intersect at a point P, then the angle between their tangents at P is
tan Φ = ± \(\frac{(d y / d x)_{1}-(d y / d x)_{2}}{1+(d y / d x)_{1}(d y / d x)_{2}}\)
The other angle of intersection will be (180° – Φ).
If two curves intersect orthogonally i.e. at right angle then \(\left(\frac{d y}{d x}\right)_{1} \cdot\left(\frac{d y}{d x}\right)_{2}\) = 1

9. Length of Tangent, Normal, Subtangent and Sub normal

• Length of Tangent = PQ = y cosec ψ = y\(\frac{\sqrt{1+(d y / d x)^{2}}}{(d y / d x)}\)
• Length of the Normal = PR = y sec ψ = y\(\sqrt{1+(d y / d x)^{2}}\)
• Length of Sub tangent = QM = y cot ψ = y /\(\left(\frac{d y}{d x}\right)\)
• Length of Sub normal = MR = y tan ψ = y\(\left(\frac{d y}{d x}\right)\)

10. Point of inflexion
If at any point P, the curve is concave on one side and convex on other side with respect to x-axis, then the point P is called the point of inflexion. Thus P is a point of inflexion if at P,
\(\frac{d^{2} y}{d x^{2}}\) = 0, but \(\frac{d^{3} y}{d x^{3}}\) ≠ 0
Also point P is a point of inflexion if f”(x) = f”‘(x) = ……… = f^n-1(x) = 0 and fⁿ(x) ≠ 0 for odd n.

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