Multiple choice

Let $\alpha ,\beta $ be the roots of $a{ x }^{ 2 }+bx+c=0;\gamma ,\delta $ be the roots of $ p{ x }^{ 2 }+qx+r=0$ and ${ D }{ 1 },{ D }{ 2 }$ are the respective discriminants of these equations. If the $\alpha ,\beta ,\gamma ,\delta $ are in AP, then ${ D }{ 1 }:{ D }{ 2 }$ is equal to

  1. $\cfrac { { a }^{ 2 } }{ { b }^{ 2 } } $
  2. $\cfrac { { a }^{ 2 } }{ { p }^{ 2 } } $
  3. $\cfrac { { b }^{ 2 } }{ { q }^{ 2 } } $
  4. $\cfrac { { c }^{ 2 } }{ { r }^{ 2 } } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If roots are in AP, let them be a-3d, a-d, a+d, a+3d. The discriminants D1 = b^2 - 4ac and D2 = q^2 - 4pr relate to the squared differences of roots. Using properties of AP and quadratic coefficients, the ratio simplifies to a^2/p^2.

AI explanation

The discriminant of a quadratic equation is the square of the difference of its roots, meaning D1 = a^2(alpha - beta)^2 and D2 = p^2(gamma - delta)^2. Since alpha, beta, gamma, and delta form an arithmetic progression, the common difference is constant, making the difference between the first two terms exactly the same as the difference between the last two terms. Therefore, alpha - beta = gamma - delta (ignoring sign since we square them). Taking the ratio of D1 to D2, the identical squared differences cancel out, leaving D1 : D2 = a^2 : p^2.

Keep practicing — related questions