# Todai Entrance Exam: Math 2015 – Problem 2

**(1)**

Proof for foci of a hyperbola is similar, with few notes:

Let :

**(2)**

It’s worth noting that:

So:

So the statement follows.

**(3)**

Similar to question (2)

**(4)**

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:

**(5)**

I’m not sure if the equation in (4) is correct so I would rather not solve this.