Let y = f (x) and y = g (x) be two curves intersecting at a point P(x₀, y₀) . Then the angle of intersection of two curves is defined as the angle between the tangent to the two curves at the point of intersection.

Let PT₁ and PT₂ be tangents to the curve y = f (x) and y = g (x) at their point of intersection.

Let Θ be the angle between two tangents PT₁ and PT₂, Θ₁ and Θ₂ are angles made by tangents PT₁ and PT₂ with the positive direction of x-axis, then

$\\m_1=\tan\theta_1=\left (\frac{d}{dx}\left ( f(x) \right ) \right )_{(x_0,y_0)}\\\\m_2=\tan\theta_2=\left (\frac{d}{dx}\left ( g(x) \right ) \right )_{(x_0,y_0)}\\\\\text{from the figure}\;\;\theta=\theta_1-\theta_2\\\\\Rightarrow \tan\theta=\tan\left (\theta_1-\theta_2 \right )=\frac{\tan\theta_1-\tan\theta_2}{1+\tan\theta_1\cdot\tan\theta_2}\\\\\\\Rightarrow \mathbf{\tan\theta=\left | \frac{\left (\frac{d}{dx}\left ( f(x) \right ) \right )_{(x_0,y_0)}-\left (\frac{d}{dx}\left ( g(x) \right ) \right )_{(x_0,y_0)}}{1+\left (\frac{d}{dx}\left ( f(x) \right ) \right )_{(x_0,y_0)}\cdot \left (\frac{d}{dx}\left ( g(x) \right ) \right )_{(x_0,y_0)}} \right |}$

Orthogonal Curves

If the angle of the intersection of two curves is a right angle then two curves are called orthogonal curves.

$\\\text{In this case,}\;\;\tan\theta=90^{\circ}\\\\\Rightarrow \left (\frac{d}{dx}\left ( f(x) \right ) \right )_{(x_0,y_0)}\cdot \left (\frac{d}{dx}\left ( g(x) \right ) \right )_{(x_0,y_0)}=-1\\\\\text{this is also the condition for two curves to be orthogonal.}$

Condition for two curves to touch each other

$\left (\frac{d}{dx}\left ( f(x) \right ) \right )_{(x_0,y_0)}= \left (\frac{d}{dx}\left ( g(x) \right ) \right )_{(x_0,y_0)}$

Study other Related Concepts

Angle of Intersection between Two Curves Current Topic

Limit

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Limit of Indeterminate Form and Algebraic limit

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Limit Using Expansion

Sandwich Theorem

Limits of Trigonometric Functions

Exponential and Logarithmic Limits

Limits of the form (1 power infinity)

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