Proof for foci of a hyperbola is similar, with few notes:
It’s worth noting that:
So the statement follows.
Similar to question (2)
Let be a point on an ellipse in . If a line is tangent to the ellipse at the point , then it has slope , and the line perpendicular to the ellipse at that point has slope . It is given from that:
I’m not sure if the equation in (4) is correct so I would rather not solve this.