Question

x²+5y²+9z²-2y(2x+3z) =0 then x:y:z=?

Answer 1

Answer:

x:y:z = 6:3:1

Step-by-step explanation:

The given equation is

${x}^{2} + 5 {y}^{2} + 9 {z}^{2} - 2y(2x + 3z) = 0$

${x}^{2} + 5 {y}^{2} + 9 {z}^{2} - 4xy - 6xz = 0$

$( {x}^{2} - 4xy + 4 {y}^{2} ) + ( {y}^{2} - 6xz + 9 {z}^{2} ) = 0$

${(x - 2y)}^{2} + {(y - 3z)}^{2} = 0$

$since \: \: {(x - 2y)}^{2} \geqslant 0$

$and \: {(y - 3z)}^{2} \geqslant 0$

the above equation is possible only if both terms are zero.

${(x - 2y)}^{2} =\: {(y - 3z)}^{2} = 0$

$(x - 2y) = (y - 3z) = 0$

$x = 2y \: \: and \: y = 3z$

$\frac{x}{2} = y = 3z$

$\frac{x}{6} = \frac{y}{3} = \frac{z}{1}$

Hence x:y:z = 6:3:1