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If a sec theta +b tan theta +c =0 and p ...

If `a sec theta +b tan theta +c =0` and `p sec theta + q tan theta + r =0` prove that `(br-qc)^2-(pc-ar)^2=(aq-bp)^2`

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`asectheta + btantheta + c = 0->(1)`
`psectheta+qtantheta +r = 0->(2)`
Now, multiplying (1) with `p` and multiplying (2) with `a` and subtracting (2) from (1),
`apsec theta +bp tan theta +cp -apsec theta -aq tantheta - ar = 0`
`=>tan theta = (ar-pc)/(bp-aq) = (pc-ar)/(aq - bp)`
Putting value of `tan theta` in (1),
`a sec theta +b((pc-ar)/(aq - bp)) = -c`
`=> asec theta = (abr - acq)/(aq-bp)`
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