Triangle in Rectangular Cartesian Coordinates

Q. Let $A(1,0),B(6,2)$, $C(\frac{3}{2},6)$ be the vertices of a triangle $ABC$. If $P$ is a point inside the triangle $ABC$ such that the triangles $APC$, $APB$ and $BPC$ have equal areas, then the length of the line segment $PQ$, where $Q$ is the point $(-\frac{7}{6},-\frac{1}{3})$, is

Q. Consider a triangle $ABC$ whose one side lies on the line $x+y=4$. If the coordinates of the orthocentre and the centroid are $(0,0)$ and $(2,4)$ respectively, and $R$ is the circumradius, then the value of $3R^2$ is

Q. Let $A(0,1),B(1,1),C(1,-1)$ and $D(-1,0)$ be four points. If $P$ is any other point, then the minimum value of $PA+PB+PC+PD$ is equal to

1. $\sqrt{5}$
2. $2\sqrt{5}$
3. $3\sqrt{5}$
4. $4\sqrt{5}$

Q. Let $A(h,k),B(1,1)$ and $C(2,1)$ be the vertices of a right angled triangle with $AC$ as its hypotenuse. If the area of the triangle is 1 square unit, then the set of values which ${}^{\prime }{k}^{\prime }$ can take is given by

1. {$-1,3$}
2. {$-3,-2$}
3. {$1,3$}
4. {$0,2$}