Q. 254.0( 9 Votes )


If x = p sec θ +

Answer :


x = p sec θ + q tan θ and y = p tan θ + q sec θ
L.H.S. = x² - y²

= (p sec θ + q tan θ) 2 – (p tan θ + q sec θ) 2
= p 2 sec 2 θ + 2pq sec θ tan θ + q 2 tan 2 θ - (p 2 tan 2 θ + 2pq tan θ sec θ + q 2 sec 2 θ )


= p 2 sec 2 θ + 2pq sec θ tan θ + q 2 tan 2 θ - p 2 tan 2 θ - 2pq tan θ sec θ -q 2 sec 2 θ

= (p 2 - q 2 ) sec 2 θ + (q 2 - p 2 ) tan 2 θ
= (p 2 - q 2 ) sec 2 θ - (p 2 - q 2 ) tan 2 θ

= (p 2 - q 2 ) (sec 2 θ - tan 2 θ)
= (p 2 – q 2 ) [Since 1 + tan 2 θ = sec 2 θ ]
= R.H.S.

x² - y² = p 2 – q 2.
Hence, proved.

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