If the ratio of the distances of a variable point $P$ from the point $(1, 1)$ and the line $x-y+2=0$ is $1: \sqrt{2}$,then the equation of the locus of $P$ is

If $A(-a, 0)$ and $B(a, 0)$ are two fixed points,then the locus of the point $P(x, y)$ on which the line segment $AB$ subtends a right angle is:

$A$ 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.

If the locus of the point,whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5:4$,is $ax^2+by^2+cxy+dx+ey+170=0$,then the value of $a^2+2b+3c+4d+e$ is equal to:

Three distinct points $A, B$,and $C$ are given in a two-dimensional coordinate plane such that for each point,the ratio of its distance from $(1, 0)$ to its distance from $(-1, 0)$ is equal to $\frac{1}{3}$. What is the circumcenter of triangle $ABC$?

Let $ABCD$ be a square. An arc of a circle with $A$ as center and $AB$ as radius is drawn inside the square joining the points $B$ and $D$. Points $P$ on $AB$,$S$ on $AD$,$Q$ and $R$ on $\operatorname{arc} BD$ are taken such that $PQRS$ is a square. Further suppose that $PQ$ and $RS$ are parallel to $AC$. Then,$\frac{\text{Area}(PQRS)}{\text{Area}(ABCD)}$ is