Find the condition that one root of $ax^2+bx+c=0$ be the reciprocal of a root of $a_1x^2+b_1x+c_1=0$.
My Attempt:
Let $x_1$ and $x_2$ be the roots of the equation $ax^2+bx+c=0$. Then,
$$x_1=\dfrac {-b+\sqrt {b^2-4ac}}{2a}$$
$$x_2=\dfrac {-b-\sqrt {b^2-4ac}}{2a}$$
Again, Let $x'_1$ and $x'_2$ be the roots of $a_1x^2+b_1x+c_1=0$. Then,
$$x'_1=\dfrac {-b_1+\sqrt {{b_1}^2-4a_1c_1}}{2a_1}$$
$$x'_2=\dfrac {-b_2+\sqrt {{b_2}^2-4a_2c_2}}{2a_2}$$
How do I proceed further?
Best Answer
Hint: the reciprocal of a root of $a_1x^2+b_1x+c_1=0$ satisfies the equation $c_1x^2+b_1x+a_1=0\,$. Then the problem reduces to finding the condition for the two equations to have a common root:
$$ \begin{cases} \begin{align} ax^2+bx+c &= 0 \\ c_1x^2+b_1x+a_1 &= 0 \end{align} \end{cases} $$
Multiplying the first equation by $a_1\,$, the second one by $c\,$, and subtracting the two in order to eliminate the constant term, then discarding the root $x=0$ which is not eligible, gives:
$$ x = \frac{b a_1 - b_1 c}{c c_1 - a a_1} $$
Substituting this expression for $x$ in either equation gives the relation between the coefficients.