Question

If a, b, c are in G.P., then the equations ax² + 2bx + c = 0 and dx² + 2ex + f = 0 have a common root if d/a, e/b, f/c are in
(a) AP
(b) GP
(c) HP
(d) none of these​

Answer 1

Answer:

If a, b, c are in G.P., then the equations ax² + 2bx + c = 0 and dx² + 2ex + f = 0 have a common root if d/a, e/b, f/c are in

$\rm\underline\bold{a)\:Ap \red{\huge{\checkmark}}}$

(b) GP

(c) HP

(d) none of these

Explanation

Given a, b, c are in GP

⇒ b² = ac

⇒ b² – ac = 0

So, ax² + 2bx + c = 0 have equal roots.

Now D = 4b² – 4ac

and the root is -2b/2a = -b/a

So -b/a is the common root.

Now,

dx² + 2ex + f = 0

⇒ d(-b/a)² + 2e×(-b/a) + f = 0

⇒ db2 /a² – 2be/a + f = 0

⇒ d×ac /a² – 2be/a + f = 0

⇒ dc/a – 2be/a + f = 0

⇒ d/a – 2be/ac + f/c = 0

⇒ d/a + f/c = 2be/ac

⇒ d/a + f/c = 2be/b²

⇒ d/a + f/c = 2e/b

⇒ d/a, e/b, f/c are in AP