At any point of a curve $\sqrt{\frac{\text {subnormal}}{\text {subtangent}}}$ is equal to

slope of the tangent at that point

We have,

Sub normal = $y \frac{d y}{d x}$, Sub-tangent = $\frac{y}{\frac{d y}{d x}}$

$\Rightarrow \sqrt{\frac{\text {Sub normal}}{\text {Sub tangent}}}=\sqrt{\frac{y \frac{d y}{d x}}{\frac{y}{\frac{d y}{d x}}}} = \frac{d y}{d x}$ = Slope of the tangent