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Asked by araima2001 | 15th Feb, 2017, 09:55: AM

Expert Answer:

Q: The line L has intercepts a and b on the coordinate axes. Keeping the origin fixed, the coordinate axes are rotated through a fixed angle. The line L has now intercepts p and q on the rotated axes. Then 
(A) a2 + b2 = p2 + q2                (B) 1/a2 + 1/b2 = 1/p2 + 1/q2
(C) a2 + p2 = b2 + q2                (B) 1/a2 + 1/p2 = 1/b2 + 1/q2
 
Solution:
begin mathsize 16px style In space the space first space coordinate space system comma space intercepts space are space straight a space and space straight b
So comma space straight x over straight a plus straight y over straight b equals 1 space...... left parenthesis straight i right parenthesis
In space the space second space coordinate space system comma space intercepts space are space straight p space and space straight q
So comma space fraction numerator straight x apostrophe over denominator straight p end fraction plus fraction numerator straight y apostrophe over denominator straight q end fraction equals 1 space space space space
where space space space space straight x apostrophe equals straight x space cos space straight theta plus straight y space sin space straight theta space and space straight y apostrophe equals negative straight x space sin space straight theta plus straight y space cos space straight theta space space space space space space.... left parenthesis straight theta space being space the space angle space of space rotation right parenthesis
rightwards double arrow straight x space open square brackets fraction numerator cos space straight theta over denominator straight p end fraction minus fraction numerator space sin space straight theta over denominator straight q end fraction close square brackets plus straight y space open square brackets fraction numerator sin space straight theta over denominator straight p end fraction minus fraction numerator cos space straight theta over denominator straight q end fraction close square brackets equals 1 space..... left parenthesis ii right parenthesis
Comparing space with space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space get
fraction numerator open square brackets fraction numerator cos space straight theta over denominator straight p end fraction minus fraction numerator space sin space straight theta over denominator straight q end fraction close square brackets over denominator 1 over straight a end fraction equals fraction numerator open square brackets fraction numerator sin space straight theta over denominator straight p end fraction plus fraction numerator space cos space straight theta over denominator straight q end fraction close square brackets over denominator 1 over straight b end fraction equals 1 over 1
rightwards double arrow fraction numerator cos space straight theta over denominator straight p end fraction minus fraction numerator space sin space straight theta over denominator straight q end fraction equals 1 over straight a space..... left parenthesis iii right parenthesis
and space fraction numerator sin space straight theta over denominator straight p end fraction plus fraction numerator space cos space straight theta over denominator straight q end fraction equals 1 over straight b..... left parenthesis iv right parenthesis
Squaring space left parenthesis iii right parenthesis space we space get comma
fraction numerator cos squared space straight theta over denominator straight p squared space end fraction plus fraction numerator space sin squared space space straight theta over denominator straight q squared space end fraction minus fraction numerator space 2 sin space straight theta space cos space straight theta over denominator pq end fraction equals fraction numerator 1 over denominator straight a squared space end fraction..... left parenthesis straight v right parenthesis
Squaring space left parenthesis iv right parenthesis space we space get comma
fraction numerator sin squared space space straight theta over denominator straight p squared space end fraction plus fraction numerator space cos squared space space straight theta over denominator straight q squared space end fraction plus fraction numerator space 2 sin space straight theta space cos space straight theta over denominator pq end fraction equals fraction numerator 1 over denominator straight b squared space end fraction..... left parenthesis vi right parenthesis
Adding space left parenthesis straight v right parenthesis space and space left parenthesis vi right parenthesis comma space we space straight g
fraction numerator cos squared space straight theta over denominator straight p squared space end fraction plus fraction numerator space sin squared space space straight theta over denominator straight q squared space end fraction plus fraction numerator sin squared space space straight theta over denominator straight p squared space end fraction plus fraction numerator space cos squared space space straight theta over denominator straight q squared space end fraction equals fraction numerator 1 over denominator straight a squared space end fraction plus fraction numerator 1 over denominator straight b squared space end fraction
rightwards double arrow fraction numerator cos squared space straight theta over denominator straight p squared space end fraction plus fraction numerator sin squared space space straight theta over denominator straight p squared space end fraction plus fraction numerator space sin squared space space straight theta over denominator straight q squared space end fraction plus fraction numerator space cos squared space space straight theta over denominator straight q squared space end fraction equals fraction numerator 1 over denominator straight a squared space end fraction plus fraction numerator 1 over denominator straight b squared space end fraction
rightwards double arrow fraction numerator 1 over denominator straight p squared space end fraction open parentheses cos squared space straight theta plus sin squared space space straight theta close parentheses plus fraction numerator space 1 over denominator straight q squared space end fraction open parentheses cos squared space straight theta plus sin squared space space straight theta close parentheses equals fraction numerator 1 over denominator straight a squared space end fraction plus fraction numerator 1 over denominator straight b squared space end fraction
rightwards double arrow fraction numerator 1 over denominator straight p squared space end fraction plus fraction numerator space 1 over denominator straight q squared space end fraction equals fraction numerator 1 over denominator straight a squared space end fraction plus fraction numerator 1 over denominator straight b squared space end fraction
Hence space proved. end style

Answered by Rebecca Fernandes | 15th Feb, 2017, 11:50: AM

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