To solve this matching problem, we need to analyze each statement/expression in Column II and determine the type of conic section it represents. Then we can match it with the conic names in Column I.

Analyzing Column II

Statement (P): The locus of the point ($h,k$) for which the line $hx+ky=1$ touches the circle $x^2+y^2=4$.

1. The equation of the given circle is $x^2+y^2=4$. This is a circle with its center at the origin $(0,0)$ and radius $r=2$.
2. The equation of the line is $hx+ky=1$, which can be rewritten as $hx+ky-1=0$.
3. The condition for a line to be tangent to a circle is that the perpendicular distance from the center of the circle to the line is equal to the radius of the circle.
4. The perpendicular distance $d$ from a point $(x_1, y_1)$ to a line $Ax+By+C=0$ is given by $d = \frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$.
5. Here, $(x_1, y_1) = (0,0)$, $A=h$, $B=k$, and $C=-1$. The radius is $r=2$.
6. So, we must have: $d = \frac{|h(0)+k(0)-1|}{\sqrt{h^2+k^2}} = 2$ $\frac{1}{\sqrt{h^2+k^2}} = 2$
7. Squaring both sides gives: $\frac{1}{h^2+k^2} = 4 \implies h^2+k^2 = \frac{1}{4}$
8. The locus of the point $(h,k)$ is $x^2+y^2 = (1/2)^2$. This is the equation of a circle.
9. Therefore, (A) $\to$ (P).

Statement (Q): Points z in the complex plane satisfying $|z+2|-|z-2|=\pm3$.

1. Let $z=x+iy$. The equation represents the locus of a point $z$ in the complex plane.
2. $|z+2|$ is the distance of the point $z$ from the point $(-2,0)$. Let's call this point $F_1$. So $PF_1 = |z-(-2)| = |z+2|$.
3. $|z-2|$ is the distance of the point $z$ from the point $(2,0)$. Let's call this point $F_2$. So $PF_2 = |z-2|$.
4. The equation is $PF_1 - PF_2 = \pm3$. This means the difference of the distances of any point on the locus from two fixed points (foci) $F_1(-2,0)$ and $F_2(2,0)$ is a constant, which is $3$.
5. This is the definition of a hyperbola.
6. The distance between foci is $2c = |2 - (-2)| = 4$, so $c=2$. The constant difference is $2a=3$, so $a=3/2$.
7. Since $c>a$ ($2 > 3/2$), it is indeed a hyperbola.
8. Therefore, (D) $\to$ (Q).

Statement (R): Points of the conic have parametric representation $x = \sqrt 3 \left( {{{1 - {t^2}} \over {1 + {t^2}}}} \right),y = {{2t} \over {1 + {t^2}}}$.

1. We can use the standard substitution $t = \tan(\theta/2)$. This gives $\cos\theta = \frac{1-t^2}{1+t^2}$ and $\sin\theta = \frac{2t}{1+t^2}$.
2. Substituting these into the given parametric equations: $x = \sqrt{3} \cos\theta \implies \frac{x}{\sqrt{3}} = \cos\theta$ $y = \sin\theta$
3. To find the Cartesian equation, we can eliminate the parameter $\theta$ using the identity $\cos^2\theta + \sin^2\theta = 1$. $\left(\frac{x}{\sqrt{3}}\right)^2 + y^2 = 1$ $\frac{x^2}{3} + \frac{y^2}{1} = 1$
4. This is the standard equation of an ellipse centered at the origin, with semi-major axis $a=\sqrt{3}$ and semi-minor axis $b=1$.
5. Therefore, (C) $\to$ (R).

Statement (S): The eccentricity of the conic lies in the interval $1 \le e < \infty$.

1. Let's recall the eccentricities ($e$) for different conic sections:
   • Circle: $e=0$
   • Ellipse: $0 < e < 1$
   • Parabola: $e=1$
   • Hyperbola: $e > 1$
2. The interval $1 \le e < \infty$ includes $e=1$ and all values greater than 1. This corresponds to both Parabola and Hyperbola.
3. Therefore, (B) $\to$ (S) and (D) $\to$ (S).

Statement (T): Points z in the complex plane satisfying ${\mathop{\rm Re}} {(z + 1)^2} = |z{|^2} + 1$.

1. Let $z = x+iy$.
2. The left-hand side (LHS) is ${\mathop{\rm Re}} {(z + 1)^2} = {\mathop{\rm Re}} {((x+1)+iy)^2}$. LHS $= {\mathop{\rm Re}} {( (x+1)^2 - y^2 + 2i(x+1)y )} = (x+1)^2 - y^2 = x^2+2x+1-y^2$.
3. The right-hand side (RHS) is $|z|^2 + 1 = |x+iy|^2 + 1 = x^2+y^2+1$.
4. Equating LHS and RHS: $x^2+2x+1-y^2 = x^2+y^2+1$ $2x - y^2 = y^2$ $2x = 2y^2$ $y^2 = x$
5. This is the standard equation of a parabola with its vertex at the origin and opening to the right.
6. Therefore, (B) $\to$ (T).

Matching Summary

• (A) Circle matches with (P).
• (B) Parabola matches with (S) and (T).
• (C) Ellipse matches with (R).
• (D) Hyperbola matches with (Q) and (S).

This gives the following mapping:

• (A) $\to$ (P)
• (B) $\to$ (S), (T)
• (C) $\to$ (R)
• (D) $\to$ (Q), (S)

Comparing with Options

• A: (A) $\to$(P); (B) $\to$(S), (T); (C) $\to$(R); (D) $\to$(R), (S) - Incorrect because (D) maps to (Q), not (R).
• B: (A) $\to$(P); (B) $\to$(S), (T); (C) $\to$(R); (D) $\to$(Q), (S) - This option correctly matches all our findings.
• C: (A) $\to$(P); (B) $\to$(S), (T); (C) $\to$(S); (D) $\to$(R), (S) - Incorrect because (C) does not map to (S) and (D) does not map to (R).
• D: (A) $\to$(P); (B) $\to$(P), (T); (C) $\to$(R); (D) $\to$(Q), (S) - Incorrect because (B) does not map to (P).

The correct option is B.