- #1

arpon

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## Homework Statement

Find the angle between the straight lines:

##(x^2+y^2)(\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta})=(x \tan{\theta} - y \sin{\alpha})^2##

## Homework Equations

[Not applicable]

## The Attempt at a Solution

Dividing by ##x^2##,

## (1+(\frac{y}{x})^2)(\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta})=(\tan{\theta} - \frac{y}{x} \sin{\alpha})^2 ##

Let, ##\frac{y}{x} =m ##.

## (1+m^2)(\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta})=(\tan{\theta} - m \sin{\alpha})^2 ##

##(\sin^2{\theta} \cos^2{\alpha}) m^2 + (2 \tan{\theta} \sin{\alpha}) m + (\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta}-\tan^2{\theta}) = 0 ##

So the solutions of this equation indicate the slopes of the two straight lines. If the solutions are ##m_1## and ##m_2##, then the angle between the two straight lines will be ##\arctan{\frac{m_1-m_2}{1+m_1 m_2}}##;

I came up with a messy equation, as I tried to calculate this. But the answer is very simple, just ##2\theta##. So, I think there is some clever technique to solve this problem.

Any suggestion will be appreciated