Question

find the angle between the following lines x cos a + y sin a = p and x sin a - y cos a =q​

Answer 1

Step-by-step explanation:

Given lines are

$\tt{ \blue{L_{1} \colon \: x \: cos( \alpha ) + y \: sin( \alpha ) = p }} \\ \tt{ \blue{L_{2} \colon \: x \: sin( \alpha ) - y \: cos( \alpha ) = q }}$

It can be rewritten in slope form, which is

$\tt{ \blue{L_{1} \colon \: y = - x \: cot( \alpha ) + p \: cosec( \alpha ) }} \\ \tt{ \blue{L_{2} \colon \: y = x \: tan( \alpha ) - q \: sec( \alpha )}} \: \: \: \: \: \: \: \:$

So, slopes of the given lines are

$\sf{ \green{m_{1} = - cot( \alpha ) \: \: \: \: \: \: \& \: \: \: \: \: \: m_{2} = tan( \alpha ) }}$

Now, Angle between the given line is given by,

$\sf{ tan( \theta) = \left| \dfrac{m_{2} -m_{1} }{1 + m_{1}m_{2} } \right| }$

$\sf{ \implies tan( \theta) = \left| \dfrac{tan( \alpha) + cot( \alpha) }{1 - 1 } \right| }$

$\sf{ \implies tan( \theta) = \left| \dfrac{tan( \alpha) + cot( \alpha) }{0 } \right| }$

$\sf{ \implies tan( \theta) = \infty }$

$\sf{ \implies \theta = \dfrac{\pi}{2} }$