$A$ circle touches the $X$-axis and also touches the circle with radius $2$ and center at $(0, 3)$. Find the locus of the center of the circle.

$A$ stick of length $r$ units slides with its ends on coordinate axes. Then the locus of the midpoint of the stick is a curve whose length is

$A$ variable straight line $AB$ divides the circumference of the circle $x^2 + y^2 = 25$ in the ratio $1 : 2$. If a tangent $CD$ is drawn to the smaller arc parallel to $AB$,such that $ABCD$ is a rectangle,then the locus of $C$ and $D$ is:

The locus of a point which moves such that the sum of the squares of its distances from the three vertices of a triangle is constant,is a circle whose centre is at the

$A$ line meets the coordinate axes at $A$ and $B$. $A$ circle is circumscribed about the triangle $OAB$. If $d_1$ and $d_2$ are the distances of the tangent to the circle at the origin $O$ from the points $A$ and $B$ respectively,then the diameter of the circle is:

The maximum possible value of $x^2+y^2-4x-6y$,where $x, y$ are real,subject to the condition $|x+y|+|x-y|=4$ is