Question

Find the of angle of insect intersection of two curves r=aø, r=a/ø
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Answer 1

Given:

two curves r=aø, r=a/ø

To Find:

Find the angle of intersection

Solution:

Before proceeding, we should know some concepts that if we differentiate an equation the first differentiation gives the slope of the equation, in here we need to find the angle of intersection of the two curves,

First, we will need to find the point of intersection for the value of x in this case the value of theta,

$r=a\theta$      -(1)

$r=\frac{a}{\theta}$        -(2)

Equating the value of r in equation 2, we have

$r=\frac{a}{\theta} \\a\theta=\frac{a}{\theta} \\\theta^2=1\\\theta=\pm1$

Now differentiating the first equation wrt theta we have,

$r=a\theta\\\frac{dr}{d\theta} =a$

So the slope of this equation at  $\theta=\pm1$ will be,

$m_1=a$

Now differentiating the second equation wrt theta we have,

$r=\frac{a}{\theta} \\\frac{dr}{d\theta}=-\frac{a}{\theta^2}$

So the slope of this equation at $\theta=\pm1$ will be,

$m_2=-a$

Now using the formula for the angle between two lines with slope m1 and m2,

$tan\theta=|\frac{m_2-m_1}{1+m_1m_2}|\\tan\theta=|\frac{-a-a}{1-a^2}|\\tan\theta= \frac{2a}{1-a^2} \\\theta=tan^{-1}(\frac{2a}{1-a^2})$

Hence, the angle of intersection between the curves is $\theta=tan^{-1}(\frac{2a}{1-a^2})$.