If $$\mathrm{P}(6,1)$$ be the orthocentre of the triangle whose vertices are $$\mathrm{A}(5,-2), \mathrm{B}(8,3)$$ and $$\mathrm{C}(\mathrm{h}, \mathrm{k})$$, then the point $$\mathrm{C}$$ lies on the circle :

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $$m$$ and $$n$$, respectively, then $$m+n^2$$ is equal to

Let the circle $$C_1: x^2+y^2-2(x+y)+1=0$$ and $$\mathrm{C_2}$$ be a circle having centre at $$(-1,0)$$ and radius 2 . If the line of the common chord of $$\mathrm{C}_1$$ and $$\mathrm{C}_2$$ intersects the $$\mathrm{y}$$-axis at the point $$\mathrm{P}$$, then the square of the distance of P from the centre of $$\mathrm{C_1}$$ is:

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies :