Angle of Intersection between Two Curves - Practice Questions & MCQ

The angle of Intersection of Two Curves

Let $y=f(x)$ and $y=g(x)$ be two curves intersecting at a point $P(x_0,y_0)$. 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 ${\mathrm{PT}}_1$ and ${\mathrm{PT}}_2$ be tangents to the curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$ and $\mathrm{y}=\mathrm{g}(\mathrm{x})$ at their point of intersection.
Let $\mathrm{\Theta }$ be the angle between two tangents ${\mathrm{PT}}_1$ and ${\mathrm{PT}}_2,{\mathrm{\Theta }}_1$ and ${\mathrm{\Theta }}_2$ are angles made by tangents ${\mathrm{PT}}_1$ and ${\mathrm{PT}}_2$ with the positive direction of x -axis, then

$\begin{array}{rl}& m_1=\tan {\theta }_1\\ & ={(\frac{d}{dx}(f(x)))}_{(x_0,y_0)}\\ & m_2=\tan {\theta }_2\\ & ={(\frac{d}{dx}(g(x)))}_{(x_0,y_0)}\end{array}$

from the figure $\theta ={\theta }_1-{\theta }_2$

$\begin{array}{rl}& \Rightarrow \tan \theta =\tan ({\theta }_1-{\theta }_2)=\frac{\tan {\theta }_1-\tan {\theta }_2}{1+\tan {\theta }_1\cdot \tan {\theta }_2}\\ & \Rightarrow \tan \theta =\left|\frac{{(\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}(\mathbf{f}(\mathbf{x})))}_{({\mathbf{x}}_{\mathbf{0}},{\mathbf{y}}_{\mathbf{0}})}-{(\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}(\mathbf{g}(\mathbf{x})))}_{({\mathbf{x}}_{\mathbf{0}},{\mathbf{y}}_{\mathbf{o}})}}{\mathbf{1}+{(\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}(\mathbf{f}(\mathbf{x})))}_{({\mathbf{x}}_{\mathbf{0}},{\mathbf{y}}_{\mathbf{o}})}\cdot {(\frac{\mathrm{d}}{\mathbf{d}\mathbf{x}}(\mathbf{g}(\mathbf{x})))}_{({\mathbf{x}}_{\mathbf{0}},{\mathbf{y}}_{\mathbf{o}})}}\right|\end{array}$

Orthogonal Curves

If the angle of the intersection of two curves is a right angle then two curves are called orthogonal curves.
In this case, $\tan \theta ={90}^{\circ }$

$\Rightarrow {(\frac{d}{dx}(f(x)))}_{(x_0,y_0)}\cdot {(\frac{d}{dx}(g(x)))}_{(x_0,y_0)}=-1$

This is also the condition for two curves to be orthogonal.
Condition for two curves to touch each other

${(\frac{d}{dx}(f(x)))}_{(x_0,y_0)}={(\frac{d}{dx}(g(x)))}_{(x0,y_0)}$