If p and q are the lengths of perpendiculars from the origin to the lines x cos θ – y sin θ = k cos 2θ and x sec θ+ y cosec θ = k, respectively, prove that p 2 + 4q 2 = k 2
The equations of given lines are
x cos θ – y sinθ = k cos 2θ … (1)
x secθ + y cosec θ= k … (2)
The
perpendicular distance (d)
of a line Ax + By + C = 0
from a point (x1,
y1)
is given by
.
On comparing equation (1) to the general equation of line i.e., Ax + By + C = 0, we obtain A = cosθ, B = –sinθ, and C = –k cos 2θ.
It is given that p is the length of the perpendicular from (0, 0) to line (1).
On comparing equation (2) to the general equation of line i.e., Ax + By + C = 0, we obtain A = secθ, B = cosecθ, and C = –k.
It is given that q is the length of the perpendicular from (0, 0) to line (2).
From (3) and (4), we have
Hence, we proved that p2 + 4q2 = k2.