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Answer

`((15a^(2))/(4))`

Explanation

Equation of any tangent to the parabola, `y^(2) = 4ax` is `y = mx + (a)/(m)`
This line will touch circle `x^(2) + y^(2) = (a^(2))/(2)`

If `((a)/(m))^(2) = (a^(2))/(2) (m^(2) + 1)`
`implies (1)/(m^(2)) = (1)/(2) (m^(2) + 1)`
`implies 2 = m^(4) + m^(2)`
`implies m^(4) + m^(2) - 2 = 0`
`implies (m^(2) - 1) (m^(2) + 2) = 0`
`implies m^(2) - 1 = 0, m^(2) = - 2`
`implies m +- 1` [`m^(2) = - 2` is not possible]
Therefore, two, common tangents are
`y = x + a` and `y = - x - a`
These two intersect at A (-a,0)
The chord of contact of A (-a, 0) for the parabola
`y^(2 = 4 ax` is 0. `y = 2a (x - a) implies x = a`
Again, length of `BC = 2 BK`
`= 2 sqrt(OB^(2) - OK^(2))`
`= 2 sqrt((a^(2))/(2) - (a^(2))/(4)) = 2 sqrt((a^(2))/(4)) = a`
and we know that DE is the latusrectum of the parabola, so its length is 4a.
Thus, area of the quadrilateral BCDE
`= (1)/(2) (BC + DE) (KL)`
`= (1)/(2) (a + 4a) ((3a)/(2)) = (15a^(2))/(4)`