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 $\alpha$.

Using distance formula, show that the points $A\left(3,1\right),B\left(6,4\right) \text{and} C\left(8,6\right)$ are collinear.

Check whether the points $\left(5,-2\right),\left(6,4\right) \text{and} \left(7,-2\right)$ are the vertices of an isosceles triangle.

Name the type of triangle formed by the points $A\left(-5,6\right),B\left(-4,-2\right) \text{and} C\left(7,5\right)$

Show that the points  $\left(a,a\right),\left(-a,-a\right)$ and $\left(-\sqrt{3}a,\sqrt{3}a\right)$ are the vertices of an equilateral triangle.