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Question

If $fraction numerator a x over denominator cos theta end fraction plus fraction numerator b y over denominator sin theta end fraction equals open parentheses a squared minus b squared close parentheses$ ,and $fraction numerator a x sin theta over denominator cos squared theta end fraction minus fraction numerator b y cos theta over denominator sin squared theta end fraction equals 0$ ,then $left parenthesis a x right parenthesis to the power of 2 divided by 3 end exponent plus left parenthesis b y right parenthesis to the power of 2 divided by 3 end exponent equals$

${(a^{2} - b^{2})}^{2 / 3}$
${(a^{2} + b^{2})}^{2 / 3}$
${(a^{2} + b^{2})}^{3 / 2}$
${(a^{2} - b^{2})}^{3 / 2}$

The correct answer is: $open parentheses a squared minus b squared close parentheses to the power of 2 divided by 3 end exponent$

$fraction numerator a x over denominator cos theta end fraction plus fraction numerator b y over denominator sin theta end fraction equals open parentheses a squared minus b squared close parentheses space space space space space..... open parentheses 1 close parentheses fraction numerator a x sin theta over denominator cos squared theta end fraction minus fraction numerator b y cos theta over denominator sin squared theta end fraction equals 0 fraction numerator a x sin theta over denominator cos squared theta end fraction equals fraction numerator b y cos theta over denominator sin squared theta end fraction a x sin cubed theta equals b y cos cubed theta therefore a x equals fraction numerator b y cos cubed theta over denominator sin cubed theta end fraction space space space space..... open parentheses 2 close parentheses S u b s t i t u t i n g space e q u a t i o n space 2 space i n space 1 comma fraction numerator b y cos cubed theta over denominator cos theta sin cubed theta end fraction plus fraction numerator b y over denominator sin theta end fraction equals open parentheses a squared minus b squared close parentheses fraction numerator b y cos squared theta over denominator sin cubed theta end fraction plus fraction numerator b y over denominator sin theta end fraction equals open parentheses a squared minus b squared close parentheses fraction numerator b y cos squared theta plus b y sin squared theta over denominator sin cubed theta end fraction equals open parentheses a squared minus b squared close parentheses fraction numerator b y open parentheses cos squared theta plus sin squared theta close parentheses over denominator sin cubed theta end fraction equals open parentheses a squared minus b squared close parentheses fraction numerator b y over denominator sin cubed theta end fraction equals open parentheses a squared minus b squared close parentheses therefore b y equals open parentheses a squared minus b squared close parentheses sin cubed theta space space space space..... open parentheses 3 close parentheses S u b s t i t u t i n g space t h e space v a l u e space o f space b y space i n space e q u a t i o n space 2 comma a x equals fraction numerator b y cos cubed theta over denominator sin cubed theta end fraction a x equals fraction numerator open parentheses a squared minus b squared close parentheses sin cubed theta cos cubed theta over denominator sin cubed theta end fraction space therefore a x equals open parentheses a squared minus b squared close parentheses cos cubed theta space space space space..... open parentheses 4 close parentheses N o w comma open parentheses a x close parentheses to the power of 2 over 3 end exponent plus open parentheses b y close parentheses to the power of 2 over 3 end exponent equals open parentheses open parentheses a squared minus b squared close parentheses cos cubed theta close parentheses to the power of 2 over 3 end exponent plus open parentheses open parentheses a squared minus b squared close parentheses sin cubed theta close parentheses to the power of 2 over 3 end exponent equals open parentheses a squared minus b squared close parentheses to the power of 2 over 3 end exponent open parentheses cos squared theta plus sin squared theta close parentheses equals open parentheses a squared minus b squared close parentheses to the power of 2 over 3 end exponent$

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From the figure AOC=

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