Under what condition the pair of straight lines represented by equation ax^2 + 2hxy + by^2 = 0 are coincident? Write it.

The pair of straight lines represented by the general second-degree equation $$a{x}^2+2hxy+b{y}^2=0$$ are coincident (i.e., they represent the same straight line) when the following condition is satisfied:

The equation represents two straight lines if the determinant of the associated matrix is zero: $$\Delta =\left|\begin{array}{cc}a& h\\ h& b\end{array}\right|=ab-h^2=0.$$

For the lines to be coincident, the additional condition is that the discriminant must also be zero: $${\Delta }^{\prime }=\left|\begin{array}{ccc}a& h& 0\\ h& b& 0\\ 0& 0& 0\end{array}\right|=0.$$ However, in the case of coincident lines, the condition simplifies to: $$\frac{a}{h}=\frac{h}{b}=\frac{2h}{a+b}.$$ This further reduces to: $$h^2=ab.$$

Thus, the necessary and sufficient condition for the pair of straight lines to be coincident is: $$\boxed{h^2=ab}.$$