1. void mouseDragged() {
     int radius =110;
     float diff = 0.31;
     float a = -atan2( center.y-mouseY, center.x- mouseX);
     p1.x = center.x + sin(a+diff)*radius;
     p1.y = center.y + cos(a+diff)*radius;
    
    
     c1.x = mouseX;
     c1.y = mouseY;
    
   }

The values for 'radius' and 'diff' are estimated, you could calculate them with angleBetween() and mag(). But this should not be done inside mouseDragged, if those vaues are not changing.

1.   PVector a1 = new PVector(center.x+p1.x, center.y+p1.y);
     PVector a2 = new PVector(center.x+c1.x, center.y+c1.y);
    
     float diff = PVector.angleBetween(a1, a2);

And it's better to calculate "diff" and "radius" only once (in the setup-function).

1. 
       PVector p1, c1, a1, a2;
       float rad, diff;
   
       void setup() {
         size(400, 400);
   
         p1 = new PVector(width/4, height/8);
         c1 = new PVector(0, height/3);
        
         rad = p1.mag();
         diff = PVector.angleBetween(p1, c1);
       }
   
       void draw() {
         background(255);
         translate(width/2, height/2);
         ellipse(0,0, 5, 5);
         ellipse(p1.x, p1.y, 5, 5);
         ellipse(c1.x, c1.y, 5, 5);
         line(p1.x, p1.y, 0, 0);
         line(c1.x, c1.y, 0, 0);
       }
   
       void mouseDragged() {     
         float a = atan2( mouseY-width/2, mouseX-height/2);
        
         p1.x =  cos(a-diff)*rad;
         p1.y =  sin(a-diff)*rad;
        
         c1.x = mouseX-width/2;
         c1.y = mouseY-height/2;    
       }

1. rad = p1.mag();

1. rad = dist(0, 0, p1.x, p1.y);

1. static public final float dist(float x1, float y1, float x2, float y2) {
       return sqrt(sq(x2-x1) + sq(y2-y1));
   }

2. static public final float mag(float a, float b) {
       return (float)Math.sqrt(a*a + b*b);
   }

mag() is about twice as fast as dist(), so always use it when you have the choice.

1. pushMatrix();
   
   translate(width/2, height/2);
   
   // draw your stuff here
   
   popMatrix();