Question

If $a$ and $b$ are the zeroes of $x^2-2x+3$, find a polynomial whose zeros are
$2a/b,2b/a$

Answer 1

Given that

$a$ and $b$ are the zeros of

$x^2-2x+3$

Then,

We know that

Sum of zeros $=-\frac{coeff.ofx}{coeff.of{x}^2}$

$a+b=-\frac{-2}{+1}$

$a+b=2--------\left(1\right)$

Now,

Product of zeros $=\frac{constantterm}{coeffof{x}^2}$

$a.b=\frac{3}{1}$

$ab=3-----------\left(2\right)$

If $\frac{2a}{b}$ and $\frac{2b}{a}$ are zeros of other polynomial

Then,

Sum of zeros $=\frac{2a^2+2b^2}{ab}$

$=\frac{2a^2+2b^2}{ab}$

$=\frac{(a^2+b^2)}{ab}$

$=\frac{2(a^2+b^2+2ab-2ab)}{ab}$

$=\frac{2(a+b{)}^2-4ab}{ab}$

$=\frac{2\times 2^2-4\times 3}{3}$

$=\frac{8-12}{3}$

Sum of zeros $=\frac{-4}{3}$

Product of zeros $=\frac{2a}{b}\times \frac{2b}{a}$

Product $=4$

Now, equation of polynomial

$x^2-$ (sum of zeros) $x+$ product $=0$

$x^2-\left(\frac{-4}{3}\right)x+4=0$

$3x^2+4x+4=0$

$3x^2+4x+12=0$

Hence, this is the answer.