Step-by-step Solution:

1.The foci are at points \((4, 3)\) and \((12, 5)\). \[\] 2.The distance between the foci is denoted as \(2ae\) (where \(a\) is the semi-major axis and \(e\) is the eccentricity). First, calculate the distance between the two foci using the distance formula: \[ \text{Distance between foci} = \sqrt{(12 - 4)^2 + (5 - 3)^2} = \sqrt{8^2 + 2^2} = \sqrt{64 + 4} = \sqrt{68}. \] So the distance between the foci is \(\sqrt{68}\). \[\] 3.The distance \(2ae\) is related to the distance between the foci, which gives: \[ 2ae = \sqrt{68}. \] \[\] 4.The relationship \(ae = \sqrt{17}\) is given, which can be used to solve for \(e\) after determining \(a\). \[\] 5.Using the formula for the total distance between the foci, we can solve for \(a\) and \(e\). From the expression \(2a = 5 + 13\), we can solve for \(a\): \[ 2a = 18 \quad \Rightarrow \quad a = 9. \] 6.Using \(ae = \sqrt{17}\) and substituting \(a = 9\): \[ e = \frac{\sqrt{17}}{a} = \frac{\sqrt{17}}{9}. \] Thus, the eccentricity \(e\) of the ellipse is \(e = \frac{\sqrt{17}}{9}\).