If $A$ and $B$ are two fixed points and $P$ is a variable point such that $PA + PB = 4$,then the locus of $P$ is a/an

The midpoint of the chord of the ellipse $x^2 + \frac{y^2}{4} = 1$ formed on the line $y = x + 1$ is

The locus of a point which moves such that the sum of its distances from two fixed points is a constant,is

The longest distance of the point $(a, 0)$ from the curve $2x^2+y^2=2x$ is

Chords of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are drawn through the positive end of the minor axis $(0, b)$. The locus of their midpoints lies on:

Let $x^2+y^2=20$ be the director circle of an ellipse $E$ whose major axis is the $X$-axis and minor axis is the $Y$-axis. If the length of the latus rectum of $E$ is $2$,then the distance between its foci is