Question

Find the slope of the line perpendicular to the line x cos alpha + y Sin Alpha = p​

Answer 1

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 = - $\dfrac{\textrm cos\alpha }{\textrm sin\alpha }$ x + $\dfrac{\textrm p}{\textrm sin\alpha }$

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 = $\frac{-1}{m}$

Or, M = $\frac{-1}{-cot\alpha }$

∴ 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