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If p and q are the lengths of the perpen...

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

Text Solution

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`"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)`
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