**Equation of Tangents
and Normal**

Let P (x₁, y₁) be any point on the curve y = f(x)

If a tangent at P makes an angle θ with the positive direction of the x – axis, then dy /dx = tanθ.

**Equation of Tangent: **Equation of a tangent at point P (x₁, y₁) is \(y-{{y}_{1}}={{\left( \frac{dy}{dx} \right)}_{({{x}_{1}},{{y}_{1}})}}(x-{{x}_{1}})\).

**Equation of Normal: **Equation of normal at point P (x₁, y₁) is \(y-{{y}_{1}}={{\left( -\frac{dx}{dy} \right)}_{({{x}_{1}},{{y}_{1}})}}(x-{{x}_{1}})\).

**Example 1:** Find the total number of parallel
tangents of f₁(x) = x² – x + 1 and f₂(x) = x³ – x² – 2x + 1.

**Solution: **f₁(x) = x² – x + 1 and f₂(x) = x³ –
x² – 2x + 1

differentiation with respect to x

f₁(x₁) = x₁² – x₁ + 1

f’₁(x₁) = 2x₁ – 1

f₂(x₂) = x₂³ – x₂² – 2x₂ + 1

f’₂(x) = 3x₂² – 2x₂ – 2

Let the tangent drawn to the curves y = f₁(x) and y = f₂(x) at (x₁, f₁(x₁)) and (x₂, f₂(x₂)) be parallel. Then

2x₁ – 1 = 3x₂² – 2x₂ – 2

2x₁ = (3x₂² – 2x₂ – 1)

which is possible for infinite numbers of ordered pairs.

So, there are infinite solution.