Circles Test 2...

A thin rod of length $$l$$ in the shape of a semicircle is pivoted at one of its ends such that it is free to oscillate in its own plane. The frequency $$f$$ of small oscillations of the semicircular rod is :

From the point $$A\left(0,3\right)$$ on the circle $${x}^{2}-4x+{\left(y-3\right)}^{2}=0$$ a chord $$AB$$ is drawn and extended to a point $$M$$ such that $$AM=2AB$$.The locus is

If $$(\alpha, \beta)$$ is a point on the circle whose centre is on the x-axis and which touches the line x + y = 0 at (2, -2),  then the greatest value of $$\alpha$$ is

Circles are drawn passing through the origin $$O$$ to intersect the coordinate axes at point $$P$$ and $$Q$$ such that $$m$$. $$PO+n.OQ=k$$, then the fixed point satisfy all of them, is given by

The centre of a circle is $$(2, -3)$$ and the circumference is $$10\pi$$. Then, the equation of the circle is

A conic $$C$$ passes through the points $$(2,4)$$ and is such that the segment of any of its tangents at any point contained between the co-ordinate axis is biscected at the point of tangency. Let $$S$$ denotes circle described on the foci $${F_1}$$ and $${F_2}$$ of the conic $$C$$ as diameter.
Equation of the circle $$S$$ is

The centre of a circle passing through the point $$(0,0),(1,0)$$ and touching the circle $$x^{2}+y^{2}=9$$ is ?

If the centroid of an equilateral triangle  $$(1,1)$$ and its one vertex is $$(-1,2)$$ , then the equation of the circumcircle is 

The equation of the circle having the lines $$x^2 + 2xy + 3x + 6y = 0$$ as its normals and having size just sufficient to contain the circle $$x (x - 4) + y(y - 3) = 0$$ is

If the equation $$\frac { \lambda ( x + 1 ) ^ { 2 } } { 3 } + \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$$ represents a circle then $$\lambda =$$