Question

If p and q are the lengths of perpendiculars from the origin to the lines x cosθ – y sinθ = k cos 2θ and x secθ+ y cosec θ = k, respectively, prove that p² + 4q² = k²

Answer 1

The equations of given lines are
x cosθ – 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