Multiple choice

The ratio of the roots of the equation $\displaystyle a_{1}x^{2} + b_{1}x + c_{1} = 0$ and $\displaystyle a_{2}x^{2} + b_{2}x + c_{2} = 0$ are equal , then

  1. $\displaystyle \left ( \frac{b_{1}}{b_{2}} \right )^{2} = \frac{a_{2} c_{2}}{a_{1} c_{1}}$
  2. $\displaystyle \left ( \frac{c_{1}}{c_{2}} \right )^{2} = \frac{b_{1} c_{2}}{c_{1} b_{2}}$
  3. $\displaystyle \left ( \frac{a_{1}}{a_{2}} \right )^{2} = \left ( \frac{c_{1}}{c_{2}} \right )^{2}$
  4. $\displaystyle \left ( \frac{b_{1}}{b_{2}} \right )^{2} = \frac{a_{1} c_{1}}{a_{2} c_{2}}$
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D Correct answer
Explanation

For a quadratic equation ax^2 + bx + c = 0, if the ratio of roots is k, then (r1/r2) = k. The condition for equal ratios of roots for two equations is (b1/b2)^2 = (a1*c1)/(a2*c2).

AI explanation

Let the common ratio of the roots be k, so the roots for the first equation are alpha and k*alpha, and for the second are beta and k*beta. Using the condition that the discriminants of both equations must have the same ratio, the squares of the differences of the roots yield ((b1)^2 - 4*a1*c1)/(a1)^2 = ((b2)^2 - 4*a2*c2)/(a2)^2. Because the ratio of roots is identical, the expression (b^2 - 4ac)/a^2 is a constant for both equations, leading to (b1/a1)^2 = (b2/a2)^2. Simplifying this proportionality relationship gives (b1/b2)^2 = a1*c1 / (a2*c2).

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