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In this paper we examine solutions in the Gaussian integers to the Diophantine equation 
$ax^{4}+by^{4}=cz^{2}$ for different choices of 
$a,b$ and 
$c$. Elliptic curve methods are used to show that these equations have a finite number of solutions or have no solution.
Let 
and let us consider a del Pezzo surface of degree one given by the equation 
. In this paper we prove that if the set of rational points on the curve Ea,b : Y2 = X3 + 135(2a−15)X−1350(5a + 2b − 26) is infinite then the set of rational points on the surface ϵf is dense in the Zariski topology.
