If the sum of the distances from a variable point $P$ to the given points $A(1,0)$ and $B(0,1)$ is $2$,then the locus of $P$ is

$A$ 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

Let $A$ be the point $(1, 2)$ and $B$ be any point on the curve $x^2 + y^2 = 16$. If the centre of the locus of the point $P$,which divides the line segment $AB$ in the ratio $3:2$ is the point $C(\alpha, \beta)$,then the length of the line segment $AC$ is

The locus of the centres of the circles,which touch the circle $x^2 + y^2 = 1$ externally,also touch the $y$-axis and lie in the first quadrant is

The tangent at any point on the curve $x^2 + y^2 = r^2$ meets the coordinate axes at $A$ and $B$. If lines are drawn through $A$ and $B$ parallel to the coordinate axes to intersect at $P$,find the locus of $P$.

The locus of the centre of a circle which passes through the point $(a, 0)$ and touches the line $x + 1 = 0$ is