If p and q are the lengths of perpendiculars from the origin to the lines `x cos theta - y sin theta = k cos 2 theta " and " x sec theta + y cosec theta = k `, respectively, prove that `p^(2) + 4q^(2) = k^(2)`.

Explanation

`"Here," P = |(-k)/(sqrt("sec"^(2)alpha + "cosec"^(2)alpha))|, p' = |(-k"cos" 2alpha)/(sqrt("cos"^(2)alpha + "sin"^(2)alpha))|`
`"Here," 4p^(2) + p'^(2) = (4k^(2))/("sec"^(2)alpha + "cosec"^(2)alpha), + (k^(2)("cos"^(2)alpha- "sin"^(2)alpha)^(2))/(1)`
`= 4k^(2)"sec"^(2)"cos"^(2)alpha + k^(2)("cos"^(4)alpha+"sin"^(4)alpha)-2k^(2)"cos"^(2)alpha xx "sin"^(2)alpha`
`=k^(2)("sin"^(2)alpha+"cos"^(2)alpha)^(2) = k^(2)`