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  4. Angle of Intersection between Two Curves - Practice Questions & MCQ

Angle of Intersection between Two Curves - Practice Questions & MCQ

Updated on Sep 18, 2023 18:34 AM

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  • Angle of Intersection of Two Curves is considered one of the most asked concept.

  • 16 Questions around this concept.

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Angle of Intersection of Two Curves

Angle of Intersection of Two Curves

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
Left-Hand Limits and Right-Hand Limits
Algebra of Limits
Limit of Indeterminate Form and Algebraic limit
Limit of Algebraic function
Limit Using Expansion
Sandwich Theorem
Limits of Trigonometric Functions
Exponential and Logarithmic Limits
Limits of the form (1 power infinity)

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