Find the point on the $x$-axis which is equidistant from the points $\left(2,-5\right)$ and $\left(-2,9\right)$.

Find a point $\mathrm{P}$ on $y$-axis which is equidistant from the points $\mathrm{A}\left(4,8\right)$ and $\left(-6,6\right)$. Also find the distance $\mathrm{AP}$.

If $\mathrm{Q}\left(0,1\right)$ is equidistant from $\mathrm{P}\left(5,-3\right)$ and $\mathrm{R}\left(x,6\right)$, find the values of $x$. Also find the distances $\mathrm{QR}$ and $\mathrm{PR}$.

Find a relation between $x$ and $y$ such that the point $\left(x,y\right)$ is equidistant from the points $\mathrm{A}\left(1,4\right)$ and $\mathrm{B}\left(-1,2\right)$.

The $x$-coordinate of a point $\mathrm{P}$ is twice its $y$-coordinate. If $\mathrm{P}$ is equidistant from the points $\mathrm{Q}\left(2,-5\right)$ and $\mathrm{R}\left(-3,6\right)$, then find the coordinates of $\mathrm{P}$.

If the points $A\left(4,3\right)$ and $B\left(x,5\right)$ are on a circle with centre $C\left(2,3\right)$, find the value of $x$. 

If the point $P\left(k-1,2\right)$ is equidistant from the points $A\left(3,k\right)$ and $B\left(k,5\right)$, find the values of $k$. 

Using distance formula, show that $\left(3,3\right)$ is the centre of the circle passing through the points $\left(6,2\right),\left(0,4\right) \text{and} \left(4,6\right)$.

The centre of a circle is $C\left(2\alpha -1,3\alpha +1\right)$ and it passes through the point $A\left(-3,-1\right)$. If a diameter of the circle is of length $20 \text{units}$, find the value(s) of $\alpha$.