Multiple choice

If the ratio of the roots of equation $\displaystyle x^{2}+px+q=0$ be equal to the ratio of the roots of $\displaystyle x^{2}+lx+m=0$ then

  1. $\displaystyle p^{2}m=q^{2}l$
  2. $\displaystyle pm^{2}=q^{2}l$
  3. $\displaystyle p^{2}l=q^{2}m$
  4. $\displaystyle p^{2}m=l^{2}q$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let roots of first be ar, br and second be cr, dr. Ratio r = a/b = c/d. Sums are -p, -l; products are q, m. (a+b)^2 / ab = (c+d)^2 / cd => p^2 / q = l^2 / m => p^2 m = l^2 q.

AI explanation

If the ratio of the roots of the first equation is equal to the ratio of the roots of the second equation, we let the first pair of roots be alpha and beta, and the second pair be alpha times k and beta times k. The condition for the first equation gives alpha squared plus beta squared divided by the product alpha beta equal to p squared divided by q, and the condition for the second equation gives k squared times (alpha squared plus beta squared) divided by k squared times alpha beta equal to l squared divided by m. Equating the two expressions yields p squared divided by q equals l squared divided by m. Cross-multiplying this relationship gives p squared m equals l squared q.

Keep practicing — related questions