Ratio of Independent Standard Normals Is Cauchy

Let $X_1, X_2 \sim \text{iid } N(0,1)$. Using the transformation $(Y, V) = (X_1/X_2,\, X_2)$:

(a) Derive the joint density $f_{Y,V}(y,v)$.

(b) Integrate out $V$ to obtain the marginal PDF of $Y = X_1/X_2$ and identify the distribution.