Mathematics · Ch 7 — Applications of Differential Calculus

Angle between Two Curves

7.2.5

Angle between Two Curves

Definition 7.3. The angle between two curves, if they intersect, is defined as the acute angle between the tangent lines to the two curves at the point of intersection.

For two lines $y=m_1x+c_1$ and $y=m_2x+c_2$ (finite slopes $m_1,m_2$ ), the acute angle $\theta$ between them satisfies

\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|.

Applied at a point of intersection of two curves, $m_1$ and $m_2$ are simply the two curves' tangent slopes there.

Note

If the two curves are parallel at the point of intersection $(x_1,y_1)$ , then $m_1=m_2$ .
If the two curves are perpendicular (cut orthogonally) at $(x_1,y_1)$ , and $m_1,m_2$ both exist and are finite, then $m_1m_2=-1$ .

Working procedure. Find the point(s) of intersection of the two curves (by solving them simultaneously), differentiate each curve to get its slope, evaluate both slopes at the point of intersection, and substitute into the $\tan\theta$ formula (or check the parallel/orthogonal special case directly). …

Figure 7.10The parabolas y = x^2 and y = (x-3)^2 intersecting at (3/2, 9/4), with their tangent lines at the intersection and the angle theta between them.

Fig. 7.10 — The parabolas y = x^2 and y = (x-3)^2 intersecting at (3/2, 9/4), with their tangent lines at the intersection and the angle theta between them.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. The parabolas y = x^2 and y = (x-3)^2 intersecting at (3/2, 9/4), with their tangent lines at the intersection and the angle theta b …

Figure 7.11The curves y = x^2 and x = y^2 intersecting at (0,0) and (1,1), with theta1 = 90 degrees at the origin and the tangent lines and angle theta2 at (1,1).

Fig. 7.11 — The curves y = x^2 and x = y^2 intersecting at (0,0) and (1,1), with theta1 = 90 degrees at the origin and the tangent lines and angle theta2 at (1,1).

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. The curves y = x^2 and x = y^2 intersecting at (0,0) and (1,1), with theta1 = 90 degrees at the origin and the tangent lines and angle the …

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