Question

If $p,q,r,s$ are rational numbers and the roots of $f\left(x\right)=0$ are eccentricities of a parabola and a rectangular hyperbola, where $f\left(x\right)=p{x}^3+q{x}^2+rx+s$, then $p+q+r+s=$

• A. $p$
• B. $-p$
• C. $2p$
• D. $0$

Answer 1

The correct option is D $0$

Given, $f\left(x\right)=p{x}^3+q{x}^2+rx+s$
And roots of this equation are eccentricities of rectangular hyperbola and parabola
We know eccentricity of parabola is $1$. Thus $1$ is a root of the given cubic equation,
$\Rightarrow f\left(1\right)=0\Rightarrow p+q+r+s=0$