Match each entry in List-I to the correct entry in List-II and...

For (P):
The radius of the circle is the perpendicular distance from $(1, 2)$ to $3x + 4y - 1 = 0$.
$r = \dfrac{|3(1) + 4(2) - 1|}{\sqrt{3^2 + 4^2}} = \dfrac{10}{5} = 2$
The equation of the circle is $(x - 1)^2 + (y - 2)^2 = 4$.
Checking the given points, $(3, 2)$ satisfies the equation.
Thus, (P) $\rightarrow$ (3).

For (Q):
The equation of a tangent to $y^2 = 8x$ with slope $m$ is $y = mx + \dfrac{2}{m}$.
Since it is also a tangent to $x^2 + y^2 = 2$, the perpendicular distance from $(0, 0)$ to the line $mx - y + \dfrac{2}{m} = 0$ is equal to the radius $\sqrt{2}$.
$\dfrac{\left|\dfrac{2}{m}\right|}{\sqrt{m^2 + 1}} = \sqrt{2}$
$\dfrac{4}{m^2} = 2(m^2 + 1)$
$m^4 + m^2 - 2 = 0$
$(m^2 + 2)(m^2 - 1) = 0$
Since $m$ is real and positive, $m = 1$.
The equation of the common tangent is $y = x + 2$.
Checking the given points, $(7, 9)$ satisfies the equation.
Thus, (Q) $\rightarrow$ (2).

For (R):
The equation of the ellipse is $\dfrac{x^2}{16} + \dfrac{y^2}{12} = 1$.
Here $a^2 = 16 and$b^2 = 12$.<br>Eccentricity$e = \sqrt{1 - \dfrac{12}{16}} = \dfrac{1}{2}$.<br>The end point of the latus rectum in the first quadrant is \left(ae, \dfrac{b^2}{a}\right) = \left(4 \times \dfrac{1}{2}, \dfrac{12}{4}\right) = (2, 3)$.
The equation of the normal at $(2, 3)$ is $\dfrac{16x}{2} - \dfrac{12y}{3} = 16 - 12$.
$8x - 4y = 4 \Rightarrow 2x - y = 1$.
Checking the given points, $(1, 1)$ satisfies the equation.
Thus, (R) $\rightarrow$ (1).

For (S):
The centre is $(0, 0)$. The focus is $(ae, 0) = (5, 0) \Rightarrow ae = 5$.
The directrix is $x = -\dfrac{16}{5} \Rightarrow \dfrac{a}{e} = \dfrac{16}{5}$.
Multiplying the two equations gives $a^2 = 16$.
Dividing the two equations gives $e^2 = \dfrac{25}{16}$.
$b^2 = a^2(e^2 - 1) = 16\left(\dfrac{25}{16} - 1\right) = 9$.
The equation of the hyperbola is $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$.
Checking the given points, $(8, 3\sqrt{3})$ satisfies the equation.
Thus, (S) $\rightarrow$ (5).

Answer: (P) $\rightarrow$ (3), (Q) $\rightarrow$ (2), (R) $\rightarrow$ (1), (S) $\rightarrow$ (5)