Positive integer solutions to $frac{1}{a} + frac{1}{b} = frac{c}{d}$

I was looking at the equation $$frac{1}{a}+frac{1}{b} = frac{c}{d},,$$ where $c$ and $d$ are positive integers such that $gcd(c,d) = 1$.

I was trying to find positive integer solutions to this equation for $a, b$, given any $c$ and $d$ that satisfy the above conditions. I was also trying to find whether there are additional requirements on $c$ and $d$ so that positive integer solutions for $a$ and $b$ can even exist.

I found that this equation simplifies to $abc – ad – bd = 0$ so that $abc = d(a+b)$.

Also, since the equation is equivalent to $a+b = ab(frac{c}{d})$, this means $a$ and $b$ are the roots of the quadratic $dx^2-abcx+abd = 0$ since their product is $ab$ and their sum is $a+b = ab(frac{c}{d})$.

However, after I analyzed the quadratic I just ended up with $a = a$ and $b = b$.

Any ideas on how to solve this further?

Again, I need to find all the conditions on the positive integers $c$ and $d$ (where $gcd(c,d) = 1$) such that positive integer solutions for $a, b$ can exist. And then also find the positive integer solutions for $a$ and $b$ given that those conditions are satisfied.