Question

$(p,q)$ is a point such that $p$ and $q$ are integers $p\ge 50$ and the equation $p{x}^2+qx+1=0$ has real roots. The square of the least distance of the point from the origin is $S$.

Find $\frac{S-225}{500}$.

Answer 1

Using distance formula,
Square of the distance of the point from the origin $=p^2+q^2$

Square of the least distance is minimum of $(p^2+q^2)$

Given quadratic equation has real roots $\Rightarrow q^2-4p\ge 0\Rightarrow q^2\ge 4p=200$

$q=15,(\because q$ is an integer and $q^2\ge 200)$

$\Rightarrow S=p^2+q^2\ge 2500+225=2725$

$\frac{S-225}{500}=\frac{2725-225}{500}=5$