# 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 θy 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.

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