23. [12] If $x$ and $y$ are real numbers such that $\frac{(x-4)^{2}}{4}+\frac{y^{2}}{9}=1$, find the largest possible value of $\frac{x^{2}}{4}+\frac{y^{2}}{9}$.
Answer: 9 The first equation is an ellipse with major axis parallel to the y-axis. If the second expression is set equal to a certain value $c$, then it is also the equation of an ellipse with major axis parallel to the y-axis; further, it is similar to the first ellipse. So the largest value of $c$ occurs when both ellipse are tangent on the x-axis, at $x=6, y=0$, which gives 9 as the largest value of $c$.