The ends of a rod of length $l$ move on two mutually perpendicular lines. The locus of the point on the rod which divides it in the ratio $1:2$ is

Through the point $(4, 5)$,a straight line is drawn making positive intercepts on the coordinate axes. The area of the triangle thus formed is least,when the ratio of the intercepts on the $X$ and $Y$ axes 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 $A(2,3), B(3,-6), C(5,-7)$ be three points. If $P$ is a point satisfying the condition $PA^2+PB^2=2PC^2$,then a point that lies on the locus of $P$ is

$A$ point $P$ on a line is at a distance of $4$ units from the origin $(0,0)$. If the line makes an angle of $60^{\circ}$ with the negative direction of the $X$-axis,then the coordinates of $P$ are

$ABC$ is a variable triangle such that $A$ is $(1, 2)$,and $B$ and $C$ lie on the line $y = x + \lambda$ (where $\lambda$ is a variable). The locus of the orthocenter of triangle $ABC$ is: