Answer:
The slope of the line perpendicular to the line x cos α + y sin α = p is tanα .
Step-by-step explanation:
Given equation of line as :
x cos α + y sin α = p
The standard equation of line is given as
y = m x + c
where m is the slope of the line
Now, for line x cos α + y sin α = p
Or, y sin α = p - x cos α
or, y sin α = - x cos α + p
Or, y = - [tex]\dfrac{\textrm cos\alpha }{\textrm sin\alpha }[/tex] x + [tex]\dfrac{\textrm p}{\textrm sin\alpha }[/tex]
Or, y = - cotα x +p cosecα
So By comparing the line, the slope of this line = m = - cotα
Now, when two lines are perpendicular then
The product of the slope of lines = - 1
Let the slope of other line = M
So, from property
m × M = - 1
∴ M = [tex]\frac{-1}{m}[/tex]
Or, M = [tex]\frac{-1}{-cot\alpha }[/tex]
∴ M = tanα
So, slope of line perpendicular to given line = M = tanα
Hence The slope of the line perpendicular to the line x cos α + y sin α = p is tanα . Answer
