Mathematics

Problem

Let

C

be the circle

x^2 + (y - 1)^2 = 2

E_1

and

E_2

be two ellipses whose centres lie at the origin and major axes lie on

x

-axis and

y

-axis respectively. Let the straight line

x + y = 3

touch the curves

C

E_1

and

E_2

P(x_1, y_1)

Q(x_2, y_2)

and

R(x_3, y_3)

respectively. Given that

P

is the mid-point of the line segment

QR

and

PQ = \frac{2\sqrt{2}}{3}

, the value of

9(x_1y_1 + x_2y_2 + x_3y_3)

is equal to _____.