Question

If a and b are the roots of x² – 3x + p = 0 and c, d are the roots x² – 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q – p) = 17 : 15.

Answer 1

Given that a and b are roots of x² – 3x + p = 0 

⇒ a + b = 3 and ab = p ...(i) 

It is given that c and d are roots of x² – 12x + q = 0 

⇒ c + d = 12 and cd = q...(ii) 

Also given that a, b, c, d are in G.P. 

Let a, b, c, d be the first four terms of a G.P.

⇒ a = a, b = ar c = ar²d = ar³

Now, 

∴a + b = 3 

⇒ a + ar = 3 

⇒ a(1 + r) = 3…(iii) 

c + d = 12 

⇒ ar² + ar³ = 12 

⇒ ar²(1 + r) = 12.....(iv) 

From (iii) and (iv) we get 

3.r² = 12 

⇒ r² = 4 

⇒ r = ±2 

Substituting the value of r in (iii) we get a = 1 

⇒ b = ar = 2

∴ c = ar² = 2² = 4 

d = ar³ = 2³ = 8 

⇒ ab = p = 2 and cd = 4×8 = 32 

⇒ q + p = 32 + 2 = 34 and q−p = 32−2 = 30

⇒ q + p:q−p = 34:30 = 17:15 

Hence, proved.