Question

If a, b, c are in G.P. 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. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

Answer: (A)

A.P.

Answer 2

b² = ac

D = 0

Ž ax² + bx + c = 0 have equal roots

a = b = – b/a

\ common root = –b/a

d$\left( \frac{–b}{a} \right)^{2}$ + 2e$\left( \frac{–b}{a} \right)$ + f = 0

db² – 2eba + a²f = 0

d(ac) – 2eba + a²f = 0

a[dc – 2eb + af] = 0

2eb = dc+af

$\frac{2e}{b}$ = $\frac{dc}{b^{2}}$ + $\frac{af}{b^{2}}$ (\ b² = ac)

$\frac{2e}{b}$ = $\frac{dc}{ac}$ + $\frac{af}{ac}$

$\frac{2e}{b}$ = $\frac{d}{a}$ + $\frac{f}{c}$ ® A.P.