Pertanyaan

Jika x = a tan α dan y cos α = b , tunjukkan bahwa: a 2 x 2 − b 2 y 2 = − 1

Jika $x = a tan α$ dan $y cos α = b$ , tunjukkan bahwa:

$\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = - 1$

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Jawaban terverifikasi

Jawaban

terbukti bahwa .

terbukti bahwa x squared over a squared minus y squared over b squared equals negative 1

Pembahasan

Ingat! Diketahui: Akan ditunjukkan bahwa Perhatikan perhitungan berikut ini: Dengan demikian terbukti bahwa .

Ingat!

$tan space alpha equals fraction numerator sin space alpha over denominator cos space alpha end fraction$

Diketahui:

$table attributes columnalign right center left columnspacing 0px end attributes row x equals cell a space tan space alpha end cell row cell y space cos space alpha end cell equals cell b rightwards arrow y equals fraction numerator b over denominator cos space alpha end fraction end cell end table$

Akan ditunjukkan bahwa

x squared over a squared minus y squared over b squared equals negative 1

Perhatikan perhitungan berikut ini:

$table attributes columnalign right center left columnspacing 0px end attributes row cell x squared over a squared minus y squared over b squared end cell equals cell open parentheses a space tan space alpha close parentheses squared over a squared minus open parentheses begin display style fraction numerator b over denominator cos space alpha end fraction end style close parentheses squared over b squared end cell row blank equals cell fraction numerator up diagonal strike a squared end strike space tan squared alpha over denominator up diagonal strike a squared end strike end fraction minus fraction numerator up diagonal strike b squared end strike over denominator cos squared alpha end fraction times fraction numerator 1 over denominator up diagonal strike b squared end strike end fraction end cell row blank equals cell tan squared alpha minus fraction numerator 1 over denominator cos squared alpha end fraction end cell row blank equals cell fraction numerator sin squared alpha over denominator cos squared alpha end fraction minus fraction numerator 1 over denominator cos squared alpha end fraction end cell row blank equals cell fraction numerator sin squared alpha minus 1 over denominator cos squared alpha end fraction end cell row blank equals cell fraction numerator negative open parentheses 1 minus sin squared alpha close parentheses over denominator cos squared alpha end fraction end cell row blank equals cell fraction numerator negative cos squared alpha over denominator cos squared alpha end fraction end cell row blank equals cell negative 1 end cell end table$

Dengan demikian terbukti bahwa x squared over a squared minus y squared over b squared equals negative 1 .