Question

if ay-bx/c=cx-az/b=bz-cy/a tgen prove tgat x/a=y/b=z=c
Do this if u can
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Answer 1

Answer:

We may obtain the following 3 equations

ay - bx = Kc ——————————(1)

cx - az = Kb ——————————(2)

bz - cy = Ka ——————————(3)

Multiplying (1) by c we obtain:

acy - bcx = Kc^2 ————————(4)

Multiplying (2) by b we obtain:

bcx - abz = Kb^2 ————————(5)

Multiplying (3) by a we obtain:

abz - acy = Ka^2 ————————(6)

AddING (4), (5), (6) we obtain:

acy - bcx + bcx - abz + abz - acy = K(a^2 + b^2 + c^2)

=> either K = 0 or/and a^2 + b^2 + c^2 =0

Now if, (a^2 + b^2 + c^2) = 0 => a, b, c are each 0 because sum of three positive number can be 0 only if each of thoese number are 0 as but we already know a, b, c can not be 0 from (Point 0)

hence, K =0 is the only deduction from above.

Substituting K=0 in (1) , (2), (3) we get:

from (1) : ay = bx => x/a = y/b ———————————(7)

from (2) : cx = az => x/a = z/c ———————————-(8)

from (3) : bz = cy => z/c = y/b ———————————-(9)

from (7), (8), (9) we see:

x/a = y/b = z/c