# Exercici 12.3

## Apartat a)

kill(all)$

Eq1:  (x-a)^2 + (y-a)^2 = 1;

depends( [y] , [x]);

diff(Eq1, x);

Eq2: (y-a)*('diff(y,x,1))+(x-a)=0;

solve ([Eq2], [a]);

a=a:(y*('diff(y,x,1))+x)/('diff(y,x,1)+1);

Eq3: (x-((y*('diff(y,x,1))+x)/('diff(y,x,1)+1)))^2 + (y-((y*('diff(y,x,1))+x)/('diff(y,x,1)+1)))^2 = 1;

ratsimp(%);

## Apartat b)

kill(all)$

Eq1:  (x-a)^2 + (y-a)^2 = R^2;

depends( [y] , [x]);

diff(Eq1, x);

Eq2: (y-a)*('diff(y,x,1))+(x-a)=0;

diff(Eq2, x);

Eq3: (y-a)*('diff(y,x,2))+('diff(y,x,1))^2+1=0;

eliminate( [Eq1, Eq2, Eq3] , [a,R]);

factor(%);

## Apartat c)

kill(all)$

Eq1:  (x-1)^2 + (y-a)^2 = 2^2;

depends( [y] , [x]);

diff(Eq1, x);

Eq2: (y-a)*('diff(y,x,1))+(x-1)=0;

solve ([Eq2], [a]);

a=a: (y*('diff(y,x,1))+x-1)/'diff(y,x,1) ;

Eq3: (x-1)^2 + (y-a)^2 = 2^2;

ratsimp(%);

## Apartat d)

kill(all)$

Eq1:  y = m*x + sqrt(1+m^2) ;

Eq1b : (y - m*x)^2 = m^2 + 1;

depends( [y] , [x]);

Eq2: diff(Eq1, x);

subst ( [m = Dy(x) ] , Eq1b) ;