Question

If ay-bx/c=cx-az/b=bz-cy/a , then prove that x/a=y/b=z/c

Answer 1

(ay-bx)/c=(cx-az)/b=(bz-cy)/a -----------(1)

let x/a =y/b=z/c is correct and it is equal to k

so,
x=ak y=bk and z=ck

now put this equation (1)
(ay-bx)/c=(abk-bak)/c=0

(cx-az)/b=(cak-ack)/b=0

(bz-cy)/a=(bck-cbk)/a=0

here we see all terms are equal hence
our assumption is correct
this is possible
(ay-bx)/c=(cx-az)/b=(bz-cy)/a
when
x/a=y/b=z/c is possible

Answer 2

Let

(ay−bx)/c=(cx−az)/b=(bz−cy)/a=λ(ay−bx)/c=(cx−az)/b=(bz−cy)/a=λ

Therefore ,

ay−bx=λc——1ay−bx=λc——1

cx−az=λb——2cx−az=λb——2

bz−cy=λa——3bz−cy=λa——3

Multiply c , b , a to 1 , 2 , 3 respectively.

Equations becomes ,

acy−bcx=λc2acy−bcx=λc2

bcx−abz=λb2bcx−abz=λb2

abz−acy=λa2abz−acy=λa2

adding equation 1 ,2 ,3 gives ,

λ∗(a2+b2+c2)=0λ∗(a2+b2+c2)=0

But ,

a2+b2+c2a2+b2+c2 is a positive quantity

Therefore,

λ=0.

Therefore ,

cx−az=0cx−az=0

bz−cy=0bz−cy=0

bz−cy=0bz−cy=0

=> x/a=y/b=z/c