Multiple choice

Let $\alpha,\beta$ be the roots of the equation $x^{2}-ax + b = 0$. If the coordinates of $A_{n}$ are $(\alpha^{n/2},\beta^{n/2})$ then,$(OA_{n+1})^{2} -a(OA_{n})^{2} + b(OA_{n-1})^{2}$ is equal to zero,O being the origin

  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

  2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

  3. Assertion is correct but Reason is incorrect

  4. Both Assertion and Reason are incorrect
Reveal answer Fill a bubble to check yourself
C Correct answer
AI explanation

The distance from the origin to point A_n is given by the square root of alpha to the power of n plus beta to the power of n. By Vieta's formulas for the equation x squared minus a times x plus b equals zero, the sum of the roots alpha plus beta equals a and the product alpha times beta equals b. Every term alpha to the power of n satisfies the recurrence relation alpha to the power of n plus 1 equals a times alpha to the power of n minus b times alpha to the power of n minus 1, and the same holds for beta. Adding these two relations proves that the distance squared term satisfies the recurrence relation, making the given expression equal to zero and the assertion correct. The provided reason is completely unrelated to this algebraic identity, so it is not the correct explanation. This leads to the result that the assertion is correct but the reason is incorrect.

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