Question

Let alpha₁,alpha₂ and beta₁,beta₂ be the roots of ax²+bx+c=0 and px²+qx+r=0 respectively. If the system alpha₁ x+alpha₂ y=0, beta₁ y+beta₂ z=0 has a non-trivial solution, then

• A. \(\dfrac{b^2}{q^2}=\dfrac{ac}{pr}\)
• B. \(\dfrac{c^2}{r^2}=\dfrac{ab}{pq}\)
• C. \(\dfrac{a^2}{p^2}=\dfrac{bc}{qr}\)
• D. None of these

Hint:

Use Vieta’s formula to simplify root-based expressions.

Answer 1

The Correct Option is A

Step 1: Non-trivial solution ⟹ determinant =0.
Step 2: Using Vieta’s relations: alpha₁alpha₂=(c)/(a), beta₁beta₂=(r)/(p)
Step 3: Condition gives: (b²)/(q²)=(ac)/(pr)