The sum of two numbers is 100, and the sum of their inverses is 2a Which is the largest of these numbers?

Let the two numbers be $x$ and $y$. We are given that $x + y = 100$ and $\frac{1}{x} + \frac{1}{y} = 2a$. We can use the quadratic formula to solve for $x$ and $y$ in terms of $a$. The quadratic formula is: ``` x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ``` where $a$, $b$, and $c$ are the coefficients of the quadratic equation. The quadratic equation in this case is: ``` x^2 + (-100 + 2a)x + 100x = 0 ``` where $a$ is the sum of the inverses of the two numbers. Substituting the values of $a$ and $b$ into the quadratic formula, we get: ``` x = \frac{100 \pm \sqrt{(100 - 2a)^2 - 4 * 100 * 100x}}{2 * 100} ``` Simplifying the expression, we get: ``` x = \frac{50 \pm \sqrt{10000 - 800a + 4a^2}}{100} ``` To find the largest value of $x$, we need to maximize the expression $\sqrt{10000 - 800a + 4a^2}$. This expression is maximized when $a = 100$, which gives us the value $x = 50$. Therefore, the largest of the two numbers is $\boxed{50}$.