Question

pls solve and send the answers in book​

Answer 1

Answer:

p² - q² = x² - y²

Step-by-step explanation:

$q\: tan \theta \:+\:p\:sec\theta\:=\:x \:.\:.\:.\:.\:.\:.(i)\\p\: tan \theta \:+\:q\:sec\theta\:=\:y \:.\:.\:.\:.\:.\:.(ii)$

• Squaring both the equations

$(q\: tan \theta \:+\:p\:sec\theta)^{2}\:=\:x^2 \:.\:.\:.\:.\:.\:.(i)\\(p\: tan \theta \:+\:q\:sec\theta)^2\:=\:y^2 \:.\:.\:.\:.\:.\:.(ii)$

• Applying the identity:-  (a + b)²  = a² + b² + 2ab

$q^2\: tan^2 \theta \:+\:p^2\:sec^2\theta\:+\:2pq\:sec\theta tan\theta=\:x^2 \:.\:.\:.\:.\:.\:.(i)\\p^2\: tan^2 \theta \:+\:q^2\:sec^2\theta\:+\:2pq\:sec\theta tan\theta=\:y^2 \:.\:.\:.\:.\:.\:.(ii)$

• Subtracting equation (ii) from equation (i)

$\implies q^2\: tan^2 \theta \:+\:p^2\:sec^2\theta\:+\:2pq\:sec\theta tan\theta=\:x^2 \:.\:.\:.\:.\:.\:.(i)\\ -(p^2\: tan^2 \theta \:+\:q^2\:sec^2\theta\:+\:2pq\:sec\theta tan\theta=\:y^2 )\:.\:.\:.\:.\:.\:.(ii)$

$\implies q^2\: tan^2 \theta \:+\:p^2\:sec^2\theta\:+\:2pq\:sec\theta tan\theta=\:x^2 \:.\:.\:.\:.\:.\:.(i)\\\:\:\:\:\:\:-\:p^2\: tan^2 \theta \:-\:q^2\:sec^2\theta\:-\:2pq\:sec\theta tan\theta=\:-\:y^2 \:.\:.\:.\:.\:.\:.(ii)\\\underline{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}$

$q^2\: tan^2 \theta -\:p^2\: tan^2 \theta\:+\:p^2\:sec^2\theta\:-\:q^2\:sec^2\theta+\:2pq\:sec\theta tan\theta-\:2pq\:sec\theta tan\theta=\:x^2 -y^2$

• Grouping so as to use the identity sec²Θ - tan²Θ = 1

$\implies\:(p^2\:sec^2\theta\:-\:p^2tan^2\theta)\: -\: (q^2\:sec^2\theta\:-\:q^2tan^2\theta) \:+ \:(\underline{2pq\:sec\theta\:tan\theta\:-\:2pq\:sec\theta\:tan\theta})\:=\:x^2-y^2$

As you can see the underlined terms get cancelled so we'll neglect them now

$\implies\:(p^2\:(sec^2\theta\:-\:tan^2\theta))\: -\: (q^2(\:sec^2\theta\:-\:tan^2\theta)) \:+ \:\:=\:x^2-y^2$

sec²Θ - tan²Θ = 1

$\implies\:(p^2\:\times\:1)\: -\: (q^2\:\times\:1)) \:=\:x^2-y^2\\\implies\:p^2\: -\: q^2 \:=\:x^2-y^2$

ANSWER :-

$\implies \mathrm {p^2\:-\:q^2\:=\:x^2\:-\:y^2}$

                         

$\mathfrak{\huge \text Hope\: this \:helps! }$