Question

If $(a{x}^2+c)y+(a^{\prime }{x}^2+c^{\prime })=0$ and $x$ is a rational function of $y$ and $ac$ is negative, then

• A. $a^{\prime }c+a{c}^{\prime }=0$
• B. $\frac{a}{a^{\prime }}=\frac{c}{c^{\prime }}$
• C. $a^2+c^2=a^{\prime 2}+c^{\prime 2}$
• D. $a{a}^{\prime }+c{c}^{\prime }=1$

Answer 1

Answer: (B)

$\frac{a}{a^{\prime }}=\frac{c}{c^{\prime }}$

Given $(a{x}^2+c)y+(a^{\prime }{x}^2+c^{\prime })=0$

$x^2(ay+a^{\prime })+(cy+c^{\prime })=0$

x is rational, $\therefore △$ must be perfect square

$△=-4(ay+a^{\prime })(cy+c^{\prime })$

$=-4(ac{y}^2+(a{c}^{\prime }+a^{\prime }c)y+a^{\prime }{c}^{\prime })$

$=-4ac(y^2+(\frac{c^{\prime }}{c}+\frac{a^{\prime }}{a})y+\frac{a^{\prime }{c}^{\prime }}{ac})$

$=-4ac\left((y+\frac{c^{\prime }}{c})(y+\frac{a^{\prime }}{a})\right)$

$△$ to be a perfect square,

$\frac{c^{\prime }}{c}=\frac{a^{\prime }}{a}$ or $\frac{c}{c^{\prime }}=\frac{a}{a^{\prime }}$

and $\left|ac\right|$ is a perfect square.