Condition for a restricted ball to be inside an ellipse

Question

Let $E\subset \mathbb{R^2}$ be the region delimited by the ellipse $\frac{x^{2}}{a^2} + \frac{y^2}{b^2} = 1$ with semi-axes $a > b$, and consider $r$ a line through the origin and $\lambda > 0$ such that $$\displaystyle\max_{y \in r}{d(y, E)} < \lambda \hspace{0.3cm} \mbox{and} \hspace{0.3cm} a> \lambda,$$ where $d(x, A)$ represents the distance between a point and a set. I would like to prove that $$r \cap B(0, \lambda) \subset E.$$ Thank you for your collaboration :)