Question

If `x/a=y/b=z/c`; calculate: `(ax+by+cz)/(a^2+b^2+c^2)- (x^2+y^2+z^2)/(ax+by+cz)` ?

If `x/a=y/b=z/c`; calculate: `(ax+by+cz)/(a^2+b^2+c^2)- (x^2+y^2+z^2)/(ax+by+cz)`

Answer 1

zero

Explanation:

`y = (bx)/a ; z = (cx)/a`

`Psi = frac{ax + (b^2x)/a + (c^2x)/a}{a^2 + b^2 + c^2} - frac{x^2 + (b^2x^2)/a^2 + (c^2x^2)/a^2}{ax + (b^2x)/a + (c^2x)/a}`

`Psi = frac{a^2x + b^2x + c^2x}{a(a^2 + b^2 + c^2)} - frac{a^2x^2 + b^2x^2 + c^2x^2}{a^3x + ab^2x + ac^2x}`

`Psi = frac{x(a^2 + b^2 + c^2)}{a(a^2 + b^2 + c^2)} - frac{x^2(a^2 + b^2 + c^2)}{ax(a^2 + b^2 + c^2)}`

`Psi = x/a - x/a = 0`

Answer 2

`0`.

Explanation:

Prerequisite :

`"If "x/a=y/b=z/c," then Each Ratio"=(lx+my+nz)/(la+mb+nc), i.e.,`

`x/a=y/b=z/c=(lx+my+nz)/(la+mb+nc)............(star)`.

Using `(star)" with "l=a, m=b, n=c`, we have,

`"Each Ratio"=(ax+by+cz)/(a^2+b^2+c^2).........(star_1)`.

Reusing `(star)`, but now with `l=x, m=y, n=z`, we get,

`"Each Ratio"=(x^2+y^2+z^2)/(ax+by+cz)...........(star_2)`.

`:." By "(star_1) and (star_2)`

`(ax+by+cz)/(a^2+b^2+c^2)=(x^2+y^2+z^2)/(ax+by+cz)`.

`rArr (ax+by+cz)/(a^2+b^2+c^2)-(x^2+y^2+z^2)/(ax+by+cz)=0`,

as Respected Vinícius Ferraz has already derived!