How to give the sketch of a set I'm asked to give a sketch of this set: $K = \{(x,y)\in\mathbb R^2: 13x^2-10xy+13y^2=72\}$ and then give the points for which the distance from the origin is maximal/ will be maximal.  Please help me. I have no idea how to solve it. Thanks in advance

The form of the equation should indicate to you that this is an ellipse with axes $45^\circ$ rotated from the standard axes. As such, the LHS can be rewritten with $(x+y)^2$ and $(x-y)^2$:
$$9(x-y)^2+4(x+y)^2=72$$
$$\frac{(x-y)^2}8+\frac{(x+y)^2}{18}=1$$
Therefore the nearest and farthest points from the origin correspond to vertices of the ellipse and have coordinates of the form $(x,\pm x)$; you should find that the semi-major and semi-minor axes have lengths $3$ and $2$ respectively, with the semi-major axis parallel to $x=y$. From there you should be able to make a sketch.
