We require to know the radius and the angle subtended by the arc from Center of each circle.

(1) Distance between the centers of two circle is 2√2.
We do not know the radius or the angle.

(2) Arcs in common portion subtend 90 degree angle at the centers of respective circles.
Angles subtended is 90.
So we can draw the figure as attached.
The line AB bisects the angle subtended from Center, so each angle is 90/2=45.
ACB = 180-45-45=90.
This triangle ACB is a right angled triangle with the two sides as r. Hypotenuse =AB=\(r\sqrt{2}\)

Combined
Now AB from statement I is \(2\sqrt{2}=r\sqrt{2}........r=2\).
We know the angle subtended from Center as 90 and radius is 2. We can find the area of common portion.

C

Let's say radius of circles are r1 and r2 and arc makes an angle of Q1 and Q2 respectively.

So, Area under intersection = Area under chord of circle having radius, r1 + Area under chord of circle having radius, r2
= {Q1*Pie*r1^2 /360 - 1/2 *r1^2} + {Q2*Pie*r2^2 /360 - 1/2 *r1^2} = (Pie/360)*(Q1*r1^2 + Q2*r2^2) - 1/2 *(r1^2 + r2^2)

Stat1: Distance between two centers = 2 Sqrt(2) = Sqrt (r1^2 + r2^2), But, we don't know, Q1 and Q2. Not sufficient.

Stat2: Q1 = Q2 = 90 degree, but (r1^2 + r2^2) is unknown. So, Not sufficient.

Combining Both stats, we have r1^2 + r2^2 and Q1= Q2 = 90 degree. Sufficient.

So, I think C.

Find the area of common portion when two circles intersect as shown below

(1) Distance between the centers of two circle is 2√2.
No info. Whether the circles have same or different radius.
Not sufficient

(2) Arcs in common portion subtend 90 degree angle at the centers of respective circles.

This statement tells that both the circles are equal but their radius can be any number.

Not sufficient

(1) + (2)

We get that both the circles are equal in size and the distance between their centers is fixed. So, the common portion can be determined.
Sufficient

Option C

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