1. The distance to a fixed point is equal to the trajectory of a fixed-length point, and the trajectory is a circle with the fixed point as the center and the fixed length as the radius.
2. The locus from a point to a point with the same distance from the two endpoints of a known line segment is the middle vertical line of this line segment.
3. The locus of points with equal distance to both sides of an angle is the bisector of the angle.
For example, because the sum of the distances from point P to two fixed points is a constant value, it means that the trajectory of point P is an ellipse, and its expression focusing on this hyperbola is x 2/a 2+y 2/a 2 =1.
The standard form of hyperbola 2x 2-3y 2 = 6 is x 2/3-y 2/2 = 1.
So c 2 = 5
So the two focal coordinates of hyperbola and ellipse are (-c, 0) and (c, 0).
Then the minimum cosine of the angle formed by the straight line formed by point P and two focal points is negative 1/9. It can be concluded that the point where the minimum value is obtained must be the intersection of the trajectory of point P and the Y axis, and the coordinates of this point where the minimum value is obtained are (0, b) and (0, -b), so it is given by cosine theorem.
2(b^2+5)-2(b^2+5)(- 1/9)=20
b=2
And can be obtained from the focal coordinates.
a^2-b^2=c^2
So a=3
So the trajectory of point P is x 2/9+y 2/4 =1.
Finally, note that X 2 represents the square of X.
I hope I can help you.