Question

How do you evaluate `\sqrt { \frac { x } { y } } + \sqrt { \frac { y } { x } } = \frac { 10} { 3}`?

Answer 1

Multiply by the product of the denominators and then square both sides.

Explanation:

Given: `sqrt(x/y)+sqrt(y/x) = 10/3`

It is implicit that both x and y must not be 0:

`sqrt(x/y)+sqrt(y/x) = 10/3; x!=0, y!=0`

Multiply both sides by `3sqrt(xy)`

`3(x+y) = 10sqrt(xy); x!=0, y!=0`

Square both sides:

`9(x^2 + 2xy+y^2) = 100xy; x!=0, y!=0`

`9x^2+18xy+9y^2 = 100xy;x!=0, y!=0`

`9x^2-92xy+9y^2 = 0;x!=0, y!=0`

The general Cartesian form for a conic section is:

`Ax^2+Bxy+Cy^2+Dx+Ey+F = 0`

But, when `F = 0`, the conic section degenerates into, a point, a line, two intersecting lines, or two parallel lines. In this case, it degenerates into two intersecting lines:

graph{9x^2-92xy+9y^2 = 0 [-10, 10, -5, 5]}

This is the graph of the last equation, above, but you can obtain the same graph by graphing `y = 9x` and `y =1/9x` except that division by 0 in the original equation prevents you from evaluating them at `x = 0`