Let the coordinate of the fixed point m be M(x, y), then there is

d 1？ =(x-y)？ /2，d2？ =(x+y)？ /2

|d 1？ -d2？ |=|(x-y)？ -(x+y)？ |/2=2

Namely. |(x-y)？ -(x+y)？ |=4

That is |xy|= 1.

This is the trajectory equation of the moving point m, and its image is two pairs of symmetrical hyperbolas.

(2) The two linear equations are: x*tanp-y=0 and x*tanp+y=0.

Let the coordinate of the fixed point m be M(x, y), then there is

d 1？ =(x*tanp-y)？ /( 1+ Tan? p)、d2？ =(x*tanp+y)？ /( 1+ Tan? p)

d 1？ +d2？ =2(x？ * Tan? p+y？ ) /( 1+ Tan? p)=6

Namely. (x？ * Tan? p+y？ ) /( 1+ Tan? p)=3

x？ /[3( 1+ 1/tan？ p)]+y？ /[3( 1+ Tan? p)]= 1

This is an elliptic equation.

Let the intersection of the tangent lines of a circle and an ellipse be a (x 1, y 1) and b (x2, y2).

Let the tangent point of the tangent on the circle be C(m, n), then there is

Is the tangent slope y' =-m/n = (y1-y2)/(x1-x2)? ( 1)

C has m on the circle o? +n？ =3 ? (2)

A and b are on the ellipse, so there are

(x 1？ * Tan? p+y 1？ ) /( 1+ Tan? p)=3 (3)

(x2？ * Tan? p+y2？ ) /( 1+ Tan? p)=3 (4)

At the same time (1)(2)(3)(4), it can be used after finishing.

x 1x2+y 1y2=0

∵OA？ =x 1？ +y 1？ ，OB？ =x2？ +y2？

AB？ =(x 1-x2)？ +(y 1-y2)？ =(x 1？ +x2？ )+(y 1？ -y2？ )-2(x 1x 2-y 1 y2)=(x 1？ +y 1？ )+(x2？ +y2？ )

∴OA？ +OB？ =AB？

That is, ∠AOB=π/2, which is a constant value and has nothing to do with P.