What is the area of the quadrilateral formed by the line $|x| + |y| = 1$?

The tangent and normal at $P(t),$ for all real positive $t,$ to the parabola $y^2 = 4ax$ meet the axis of the parabola in $T$ and $G$ respectively. Find the angle at which the tangent at $P$ to the parabola is inclined to the tangent at $P$ to the circle passing through the points $P, T,$ and $G$.

If the foci of $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ and $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{9}=1$ coincide,then the value of $a$ is

The locus of the foot of the perpendicular drawn from the centre upon any tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is

Columns $1, 2$ and $3$ contain conics,equations of tangents to the conics,and points of contact,respectively.

$Column 1$	$Column 2$	$Column 3$
$(I) x^2+y^2=a^2$	$(i) my=m^2x+a$	$(P) (a/m^2, 2a/m)$
$(II) x^2+a^2y^2=a^2$	$(ii) y=mx+a\sqrt{m^2+1}$	$(Q) (-ma/\sqrt{m^2+1}, a/\sqrt{m^2+1})$
$(III) y^2=4ax$	$(iii) y=mx+\sqrt{a^2m^2-1}$	$(R) (-a^2m/\sqrt{a^2m^2+1}, 1/\sqrt{a^2m^2+1})$
$(IV) x^2-a^2y^2=a^2$	$(iv) y=mx+\sqrt{a^2m^2+1}$	$(S) (-a^2m/\sqrt{a^2m^2-1}, -1/\sqrt{a^2m^2-1})$

$(1)$ The tangent to a suitable conic (Column $1$) at $(\sqrt{3}, 1/2)$ is $\sqrt{3}x+2y=4$. Which combination is correct?
$(2)$ If a tangent to a suitable conic (Column $1$) is $y=x+8$ and its point of contact is $(8, 16)$,which combination is correct?
$(3)$ For $a=\sqrt{2}$,if a tangent is drawn to a suitable conic (Column $1$) at $(-1, 1)$,which combination is correct?

$A$ common tangent $T$ to the curves $C_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ and $C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1$ does not pass through the fourth quadrant. If $T$ touches $C_{1}$ at $(x_{1}, y_{1})$ and $C_{2}$ at $(x_{2}, y_{2})$,then $|2x_{1} + x_{2}|$ is equal to $......$