Match the following:
Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
Then the value of $ pq $ is
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
A chemical element with the symbol Cl and the atomic number 17 is ‘Chlorine’. It is the second-lightest of the halogens that appears between fluorine and bromine in the periodic table and its properties are median between them. At room temperature, it is a yellow-green gas. Also, it has two stable isotopes, 35Cl and 37Cl. These are the only two natural isotopes occurring in quantity, with 35Cl making up 76% of natural chlorine and 37Cl making up the remaining 24%.