If the straight lines `ax+by+p=0 and x cos alpha +y sin alpha = c` enclose an angle `pi//4` between them and meet the straight line `x sin alpha - y cos alpha = 0 ` in the same point , then

Options

A. `a^(2) + b^(2) = c^(2)`
B. `a^(2) + b^(2) =c^(2)`
C. `a^(2) + b^(2) = 2c^(2)`
D. `a^(2) + b^(2) =4`

Answer

B

Explanation

The straight lines ` x cos alpha + y sin alpha= c` and `x sin alpha - y cos alpha = 0` are mutually perpendicular . Line `x cos alpha + y sin alpha =c` is in normal form.So , perpendicular from the orgin as this line is of length c and makes angle `pi//4` with OX.
The line sin `alpha - y cos alpha = 0` passes through the origin and makes angle `alpha` with x-axis. So,it is along the perpendicular from the origin on `x cos alpha + y sin alpha = c` . Line `ax + by + c=0` makes angle `pi//4` with `x cos alpha + y sin alpha =c` and `x cos alpha + y sin alpha = c` and `x sin alpha - y cos alpha =0` are mutually perpendicular. So , `ax + by + c=0` is a bisector of the angle between them.
In `Detla OPQ`, we have
`OQ = PO sin .(pi)/(4) = (c)/(sqrt(2))`
Also,
OQ= Lenght of the perpendicular from (0,0) on
`ax + by + c = 0`
`rArr OQ = (c)/(sqrt(a^(2)+b^(2)) rArr (c)/(sqrt(2)) = (c)/(sqrt(a^(2) +b^(2)) rArr a^(2) + b^(2) = 2`