Angle between two curves and condition for two curves cutting at right angle.
Understand the Problem
The question is discussing the relationship between the slopes of tangents of two curves at their intersection points and the angle between them. It outlines the formula to find the tangent of the angle and the condition for the two curves to intersect at right angles.
Answer
To find the angle $\theta$ between the tangents of two curves, use $$ \tan \theta = \pm \frac{m_1 - m_2}{1 + m_1 m_2}, $$ with orthogonality condition $m_1 m_2 = -1.$
Answer for screen readers
The angle between the tangents of two curves at their point of intersection can be calculated using the formula
$$ \tan \theta = \pm \frac{m_1 - m_2}{1 + m_1 m_2} $$
with the condition for orthogonality being
$$ m_1 m_2 = -1. $$
Steps to Solve
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Understanding the slopes of tangents
Given the slopes $m_1$ and $m_2$ of tangents at the point of intersection $P$ of two curves, these values are essential for calculating the angle between the tangents. -
Formula for the tangent of the angle
To find the tangent of the angle $\theta$ between the two tangents, use the formula:
$$ \tan \theta = \pm \frac{m_1 - m_2}{1 + m_1 m_2} $$
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Condition for orthogonality
For the two curves to intersect orthogonally (at right angles), their slopes must satisfy the condition:
$$ m_1 m_2 = -1 $$
This means if one slope is positive, the other must be negative, indicating perpendicular tangents.
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Interpretation of $\tan \theta$
If the slopes yield $\tan \theta = 0$, it implies that the angle between the tangents is $0^\circ$; therefore, they are parallel. If $\tan \theta$ is undefined, it indicates a $90^\circ$ angle between the tangents.
The angle between the tangents of two curves at their point of intersection can be calculated using the formula
$$ \tan \theta = \pm \frac{m_1 - m_2}{1 + m_1 m_2} $$
with the condition for orthogonality being
$$ m_1 m_2 = -1. $$
More Information
This relationship is fundamental in calculus and geometry, especially when studying curves and their intersections. The concept of orthogonality helps in understanding the behavior of curves around points where they intersect.
Tips
- Confusing the values of $m_1$ and $m_2$. Always ensure that you correctly identify the slopes of the tangents from their respective curves at the intersection point.
- Forgetting the condition for orthogonality. It's crucial to use the condition $m_1 m_2 = -1$ when determining whether the curves intersect at right angles.
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