If p and q are respectively the perpendiculars from the origin upon the straight lines, whose equations are ` x sec theta + y cosec theta =a and x cos theta -y sin theta = acos 2 theta , then` `4p^(2) + q^(2)` is equal to

Options

A. `5a^(2)`
B. `4a^(2)`
C. `3a^(2)`
D. `a^(2)`

Answer

D

Explanation

Since, the perpendicular distance from (0,0) to the line x ` sec theta +y cosec theta -a =0` is P.
`p = (|0+0-a|)/(sqrt(sec^(2)theta+cosec^(2)theta)) = a/(sqrt(1/cos^(2)theta + 1/sin^(2)theta))`
` (a sin theta cos theta)/(sqrt(sin^(2) theta + cos^(2) theta)) = (a sin theta cos theta)/1 .2/2 = (a sin 2theta)/2`
Also, the perpendicular distance from (0,0) to the line ` x cos theta -y sin theta- a cos 2 theta = 0` is q.
` q = (|0-0-a cos 2 theta|)/(sqrt(cos^(2) theta + sin^(2)theta)) = a cos2 theta `
Now,` 4p^(2) +q^(2) = 4 ((a sin 2theta)/2)^(2) + (a cos 2theta)^(2)`
` =a^(2) sin^(2) 2 theta + a^(2) cos^(2) 2theta = a^(2)`