If the three lines x – 3y = p, ax + 2y = q and ax + y = r form a right‑angled triangle then :

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Question

a² – 9a + 18 = 0

a² – 6a – 18 = 0

a² – 9a + 12 = 0

a² – 6a – 12 = 0

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Since three lines x – 3y = p,

ax + 2y = q and ax + y = r

$∴$ product of slopes of any two lines = – 1

Suppose ax + 2y = q and x – 3y = p are $⊥$ to each other.

$∴ \frac{- a}{2} \times \frac{1}{3} = - 1 \Rightarrow a = 6$

Now, consider option are by one a = 6 satisfies only option (a)

$∴$ Required answer is a² – 9a + 18 = 0