Problem 246
Tangents to an ellipse
A definition for an ellipse is:
Given a circle c with centre M and radius r and a point G such that d(G,M)<r, the locus of the points that are equidistant from c and G form an ellipse.
The construction of the points of the ellipse is shown below.
Given are the points M(-2000,1500) and G(8000,1500).
Given is also the circle c with centre M and radius 15000.
The locus of the points that are equidistant from G and c form an ellipse e.
From a point P outside e the two tangents t1 and t2 to the ellipse are drawn.
Let the points where t1 and t2 touch the ellipse be R and S.
For how many lattice points P is angle RPS greater than 45 degrees?
椭圆的切线
椭圆的一种定义方式为：
给定以M为圆心、半径为r的圆c，取一点G满足d(G,M)<r，则到圆c和到点G的距离相同的点的轨迹构成一个椭圆。
如下动画展示了如何构造一个椭圆。
已知点M(-2000,1500)和点G(8000,1500)。
同样已知以M为圆心、半径为15000的圆c。
到圆c和到点G的距离相同的点的轨迹构成一个椭圆e。
从椭圆e外的一个点P引椭圆的两条切线t1和t2。
记t1和t2切椭圆于点R和点S。
有多少个格点P使得角RPS大于45度？