For any real number $t$,the point $\left(\frac{8 t}{1+t^2}, \frac{4\left(1-t^2\right)}{1+t^2}\right)$ lies on a/an

The locus of the midpoints of the chords of the circle $x^2-2x+y^2=0$ drawn from the point $(0,0)$ on it is

The locus of the point of intersection of the lines,$\sqrt{2}x - y + 4\sqrt{2}k = 0$ and $\sqrt{2}kx + ky - 4\sqrt{2} = 0$ (where $k$ is any non-zero real parameter) is

Let $P$ be a point on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ and the line through $P$ parallel to the $Y$-axis meets the circle $x^{2}+y^{2}=9$ at $Q$,where $P$ and $Q$ are on the same side of the $X$-axis. If $R$ is a point on $PQ$ such that $\frac{PR}{RQ}=\frac{1}{2}$,then the locus of $R$ is

Let $T'$ be the line passing through the points $P(-2, 7)$ and $Q(2, -5)$. Let $F_1$ be the set of all pairs of circles $(S_1, S_2)$ such that $T'$ is tangent to $S_1$ at $P$ and tangent to $S_2$ at $Q$,and also such that $S_1$ and $S_2$ touch each other at a point,say,$M$. Let $E_1$ be the set representing the locus of $M$ as the pair $(S_1, S_2)$ varies in $F_1$. Let the set of all straight line segments joining a pair of distinct points of $E_1$ and passing through the point $R(1, 1)$ be $F_2$. Let $E_2$ be the set of the mid-points of the line segments in the set $F_2$. Then,which of the following statement$(s)$ is (are) $TRUE$?

Consider a rigid square $ABCD$ as in the figure with $A$ and $B$ on the $X$ and $Y$-axes,respectively. When $A$ and $B$ slide along their respective axes,the locus of $C$ forms a part of