If *p
*and *q *are the lengths of perpendiculars from the origin to
the lines *x *cos *θ* – *y
*sin *θ* = *k
*cos 2*θ* and *x *sec *θ*+
*y *cosec *θ* =
*k*, respectively, prove that *p* ^{2}
+ 4*q* ^{2} =
*k* ^{2}

The equations of given lines are

*x*
cos *θ* – *y*
sin*θ* = *k* cos
2*θ* … (1)

*x*
sec*θ* + *y* cosec
*θ*= *k* …
(2)

The
perpendicular distance (*d*)
of a line *Ax *+ *By *+ *C* = 0
from a point (*x*_{1},
*y*_{1})
is given by.

On
comparing equation (1) to the general equation of line i.e., *Ax *+
*By *+ *C* = 0, we obtain *A* = cos*θ*,
*B* = –sin*θ*,
and *C* = –*k* cos 2*θ*.

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* = sec*θ*,
*B* = cosec*θ*,
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 *p*^{2}
+ 4*q*^{2} =
*k*^{2}.

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