True and False ll

Given $\frac{1}{x}$ $-$ $\frac{1}{y}$ $=$ $\frac{1}{z}$ where $x$ , $y$ and $z$ are postive integers

On simplifying the equation

$=$ $\frac{y-x}{xy}$

Let us assume $y-x$ $=$ $1k$ for some positive integer $k$

So $xy$ $=$ $zk$

Now according to the question, we have to find whether $hxyz$ and $h (y-x)$ are perfect squares

$hxyz$ $=$ $hz$ $×$ $xy$

$=$ $hz$ $×$ $zk$

$=$ $hk$ $×$ $z^2$

Now observe $h=k$ because $1$ and $z$ are co-prime integers . Meaning $\frac{1}{z}$ is its standard form of rational number. If $\frac{1}{z}$ were not in its standard form then for it be in its standard form, both the numerator ( $1$ ) and denominator( $z$ ) have to be both divided by their g.c.d that is $k$ .

$=$ $h^2$ $×$ $z^2$ $=$ $(zh)^2$ So it is a perfect square.

Similarly for other case similar substitution can be done.