Does this metric induce the same topology as the euclidean does?

Question

I do not really know how to solve this one. Id be grateful for some tips on how to solve it

Here is the question: 1) Decide if the given Metric d in $\mathbb{R^2} $ induces the same Topology as the Euclidean metric does. 2) Does d come from a Norm in $\mathbb{R^2} $ and if so, does the norm come from a Scalar product?

Answer 1

Hint. (1) What is, for example, the open ball of radius $1$ around $(1,0)$ accourding to your metric? Is this set open according to the Euclidean topology?

(2) If the metric comes from a norm, then the translation of a ball is itself a ball.