Orthogonality of Two Circles

Q. If a circle passes through the point $(a,b)$ and cuts the circle $x^2+y^2=p^2$ orthogonally, then the equation of the locus of its centre is

1. $x^2+y^2-3ax-4by+(a^2+b^2-p^2)=0$
2. $2ax+2by-(a^2+b^2+p^2)=0$
3. $2ax+2by-(a^2-b^2+p^2)=0$
4. $x^2+y^2-2ax-3by+(a^2-b^2-p^2)=0$

Q.

If two circles which pass through the points (0, a) and (0, -a) cut each other orthogonally and touch the straight line $y=mx+c$, then

1. $c^2=a^2(1+m^2)$
   

2. $c^2=a^2(2+m^2)$
   

3. $c^2=2a^2(1+m^2)$

4. $c^2=a^2|1+m^2|$
   

Q. If $A(-2,3)B(2,-1)C(3,1)$ then the locus of $P$ such that $P{A}^2+P{B}^2+P{C}^2=7$ is a circle with centre

1. $(2,1)$
2. $(1,2)$
3. $(1,1)$
4. $(2,2)$