If a point $(x, y) = (\tan \theta + \sin \theta, \tan \theta - \sin \theta)$,then the locus of $(x, y)$ is

$A$ frog is presently located at the origin $(0,0)$ in the $XY$-plane. It always jumps from a point with integer coordinates to a point with integer coordinates,moving a distance of $5$ units in each jump. What is the minimum number of jumps required for the frog to go from $(0,0)$ to $(0,1)$?

Let $A$ be a fixed point $(0,6)$ and $B$ be a moving point $(2t, 0)$. Let $M$ be the mid-point of $AB$ and the perpendicular bisector of $AB$ meets the $y$-axis at $C$. The locus of the mid-point $P$ of $MC$ is:

The locus of the mid-points of the points of intersection of $x \cos \theta + y \sin \theta = 1$ with the coordinate axes is

Let $A=(2,3)$ and $B=(3,-5)$ be two vertices of $\triangle ABC$ such that $C$ is a point on the line $L \equiv 3x + 4y - 5 = 0$. Then the locus of the centroid of $\triangle ABC$ is a line parallel to

The locus of the mid-points of the perpendiculars drawn from points on the line $x=2y$ to the line $x=y$ is: