1. Write the tangent in slope form

A common tangent to both curves can be written as $y=mx+c.$

2. Condition for tangency to the circle

The circle is $x^2+y^2=2,$ so its center is $(0,0)$ and radius is $\sqrt{2}$.

For the line $y=mx+c$ to be tangent to the circle, the perpendicular distance from the origin to the line must be $\sqrt{2}$: $\frac{|c|}{\sqrt{1+m^2}}=\sqrt{2}.$ Squaring, $c^2=2(1+m^2).$

3. Condition for tangency to the parabola

The parabola is $y^2=8x=4ax,$ so $a=2$.

A tangent to $y^2=4ax$ with slope $m$ is $y=mx+\frac{a}{m}=mx+\frac{2}{m}.$ Hence for tangency to the parabola, $c=\frac{2}{m}.$

4. Common tangent condition

Substitute $c=\frac{2}{m}$ into the circle condition: $\left(\frac{2}{m}\right)^2=2(1+m^2).$ So, $\frac{4}{m^2}=2+2m^2.$ Multiply by $m^2$: $4=2m^2+2m^4,$ $2=m^2+m^4.$ Let $t=m^2$. Then $t^2+t-2=0,$ $(t-1)(t+2)=0.$ Since $t\ge 0$, we get $m^2=1 \implies m=\pm 1.$

Thus the two common tangents are:

• for $m=1$, $y=x+2,$
• for $m=-1$, $y=-x-2.$

5. Points of contact on the parabola

For $y^2=4ax$, the tangent $y=mx+\frac{a}{m}$ touches at $\left(\frac{a}{m^2},\frac{2a}{m}\right).$ Here $a=2$.

• For $m=1$: $R=(2,4).$
• For $m=-1$: $S=(2,-4).$

6. Points of contact on the circle

For the circle $x^2+y^2=2$, tangent at $(x_1,y_1)$ is $xx_1+yy_1=2.$

For line $y=x+2$

Rewrite as $-x+y-2=0 \implies y-x=2.$ Compare with tangent form. The point of contact is the foot of perpendicular from origin to the line.

Using formula for foot from $(0,0)$ to $ax+by+d=0$ with $a=-1,b=1,d=-2$: $\left(\frac{-ad}{a^2+b^2},\frac{-bd}{a^2+b^2}\right)=\left(\frac{-(-1)(-2)}{2},\frac{-(1)(-2)}{2}\right)=(-1,1).$ So $P=(-1,1).$

For line $y=-x-2$

Rewrite as $x+y+2=0.$ Foot from origin to this line: $Q=\left(\frac{-(1)(2)}{2},\frac{-(1)(2)}{2}\right)=(-1,-1).$ So $Q=(-1,-1).$

7. Coordinates of the quadrilateral

The vertices are $P(-1,1),\quad Q(-1,-1),\quad S(2,-4),\quad R(2,4).$

This is a trapezium with parallel sides:

• $PQ$ is vertical, length $PQ=2,$
• $RS$ is vertical, length $RS=8.$

Distance between the parallel sides is the horizontal distance: $2-(-1)=3.$

Therefore area is