Let $a_{1}, a_{2}, a_{3}$ be three distinct real numbers in geometric progression. If the equations $a_{1} x ^ 2 + 2a_{2}x + a_{3} = 0$ and $b_{1} x ^ 2 + 2b_{2}x + b_{3} = 0$ have a common…

Question

Let $a_{1}, a_{2}, a_{3}$ be three distinct real numbers in geometric progression. If the equations $a_{1} x ^ 2 + 2a_{2}x + a_{3} = 0$ and $b_{1} x ^ 2 + 2b_{2}x + b_{3} = 0$ have a common root, then which of the following is necessarily true?

Accepted Answer

D - $\dfrac{b_{1}}{a_{1}}, \dfrac{b_{2}}{a_{2}}, \dfrac{b_{3}}{a_{3}}$ are in arithmetic progression. a2^2=a1a3 makes the first equation a perfect square with double root x=-a2/a1. Substituting into the second: b1 a2^2/a1^2 - 2b2 a2/a1 + b3 = 0; dividing by a2^2/a1 = a3 gives b1/a1 + b3/a3 = 2 b2/a2, so b1/a1, b2/a2, b3/a3 are in AP.

Answer

A - $\dfrac{b_{1}}{a_{1}}, \dfrac{b_{2}}{a_{2}}, \dfrac{b_{3}}{a_{3}}$ are in geometric progression

Answer

B - $b_{1}, b_{2}, b_{3}$ are in geometric progression

Answer

C - $b_{1}, b_{2}, b_{3}$ are in arithmetic progression

Let \(a_{1}, a_{2}, a_{3}\) be three distinct real numbers in geometric progression. If the equations \(a_{1} x ^ 2 + 2a_{2}x + a_{3} = 0\) and \(b_{1} x ^ 2 + 2b_{2}x + b_{3} = 0\) have a common…

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