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 :

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point $$(3,2)$$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point $$(5,5)$$ is :

Let $$\mathrm{C}$$ be a circle with radius $$\sqrt{10}$$ units and centre at the origin. Let the line $$x+y=2$$ intersects the circle $$\mathrm{C}$$ at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$. Let $$\mathrm{MN}$$ be a chord of $$\mathrm{C}$$ of length 2 unit and slope $$-1$$. Then, a distance (in units) between the chord PQ and the chord $$\mathrm{MN}$$ is

A square is inscribed in the circle $$x^2+y^2-10 x-6 y+30=0$$. One side of this square is parallel to $$y=x+3$$. If $$\left(x_i, y_i\right)$$ are the vertices of the square, then $$\Sigma\left(x_i^2+y_i^2\right)$$ is equal to: