If one of the lines given by the equation 2𝑥2 + axy + 3y2 = 0 coincides with one of those given by 2x2+ b𝑥𝑦 - 3𝑦2 = 0 and the other lines represented by them are perpendicular then 𝑎2 + 𝑏2 =
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 + 𝑏2 = 25 + 1 = 26
Hence, 26.
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