Question

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

Answer 1

The length of the perpendicular from the origin to the lines xcosθ−ysinθ=kcos2θ and xsecθ+y cosecθ=k are p and q respectively.

The general form of the equation of line is given by,

Ax+By+C=0 (1)

Rearrange the terms of equation of line xcosθ−ysinθ=kcos2θ .

xcosθ−ysinθ−kcos2θ=0

Compare the above expression with the general form of equation of line from equation (1).

A=cosθ, B=−sinθ, C=−kcos2θ (2)

The formula for the perpendicular distance d of a line Ax+By+C=0 from a point ( x 1 , y 1 ) is given by,

d= | A x 1 +B y 1 +C | A 2 +B 2 (3)

Substitute the value of ( x 1 , y 1 ) as ( 0,0 ) and the values of A , B ,and C from equation (2) to equation (3).

p= | A( 0 )+B( 0 )+C | A 2 +B 2 = | C | A 2 +B 2 = | −kcos2θ | cos 2 θ+ sin 2 θ =| −kcos2θ | (4)

Rearrange the terms of equation of line xsecθ+y cosecθ=k .

xsecθ+y cosecθ+( −k )=0

Compare the above expression with the general form of equation of line from equation (1).

A=secθ, B=cosecθ, C=−k (5)

Substitute the value of ( x 1 , y 1 ) as ( 0,0 ) and the values of A , B ,and C from equation (5) to equation (3).

p= | A( 0 )+B( 0 )+C | A 2 +B 2 = | C | A 2 +B 2 = | −k | sec 2 θ+ cosec 2 θ (6)

As per the question L.H.S. of the given expression is,

p 2 +4 q 2

Substitute the values of p and q from equation (4) and equation (6) in the above expression.

p 2 +4 q 2 = ( | −kcos2θ | ) 2 +4 ( | | −k | sec 2 θ+ cosec 2 θ | ) 2 = k 2 cos 2 2θ+ 4 k 2 sec 2 θ+ cosec 2 θ = k 2 cos 2 2θ+ 4 k 2 ( 1 cos 2 θ + 1 sin 2 θ ) = k 2 cos 2 2θ+ 4 k 2 ( cos 2 θ+ sin 2 θ cos 2 θ sin 2 θ )

Further simplify the above expression

p 2 +4 q 2 = k 2 cos 2 2θ+ 4 k 2 ( 1 cos 2 θ sin 2 θ ) = k 2 cos 2 2θ+4 k 2 cos 2 θ sin 2 θ = k 2 cos 2 2θ+ k 2 ( 2cosθsinθ ) 2 = k 2 cos 2 2θ+ k 2 ( sin2θ ) 2

Further simplify the above expression

p 2 +4 q 2 = k 2 cos 2 2θ+ k 2 sin 2 2θ = k 2 ( cos 2 2θ+ sin 2 2θ ) = k 2

Hence p 2 +4 q 2 = k 2 .