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# CBSE Class 10 Answered

If a secA + b tanA+ c =0 and                       psecA+qsecA+r=0 prove that                     (br-qc)^2-(pc-ar)^2=(aq-bp)^2 Please answer in a simple way
Asked by guptaabhinav0809 | 10 Feb, 2019, 11:51: PM
For convenience, let us use the substitutions,  sec A = S   and tan A = T

given equations are written as :

a×S + b×T  = - c   ...................(1)
p×S + q×T  = - r   ...................(2)

by multiplying (1) by p  and by multiplying (2) by a, we rewrite the equations as follows

a×p×S + b×p×T = - pc  ...........................(3)
a×p×S + q×a×T = - ar  ..........................(4)

subtracting (4) from (3), we get,  ( b×p - q×a ) T = (ar - pc)   or T = tan A = ( pc - ar) / (aq - pb)  ............................(5)

Similarly by multiplying (1) by q  and by multiplying (2) by b and eliminating T,  we get, S = sec A = ( br - cq )/(aq - pb).................(6)

we have, sec2A - tan2A  = 1  ..............(7)

by substituting for secA and tanA from (5) and (6) in eqn.(7), it can be shown that,  (br-qc)2 - (pc-ar)2 = (aq-pb)2
Answered by Thiyagarajan K | 11 Feb, 2019, 10:41: AM

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