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Explanation:

The two given equations contain 3 lines.

Let the slope of common line between 2 equations be 'm'
Let the slope of remaining two perpendicular lines be b and -1/n.

Given, 2𝑥2 + axy + 3y2 = 0
⇒ 23x2 + a3xy + y2 = (y - mx)(y - nx)
∴ m + n = a3 ...(1) and mn = 23   ...(2)

Also, 2𝑥2 + bxy - 3y2 = 0
⇒ -23x2 - b3xy + y2 = y-mxy+1nx
∴ - m + 1n = -b3   ...(3)  and -mn = -23   ...(4)

Multiplying (2) and (4)
⇒ m2 = 4/9

Case 1: m = + 2/3 ⇒ n = 1
⇒ a/3 = m + n ⇒ a = 5
⇒ - b/3 = - m + 1/n ⇒ b = 1

Case 2: m = - 2/3 ⇒ n = - 1
⇒ a/3 = m + n ⇒ a = - 5
⇒ -b/3 = - m + 1/n ⇒ b = - 1

In both cases: 𝑎2 + 𝑏= 25 + 1 = 26

Hence, 26.

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