In $\triangle ABC$,if $A$ is $(1,2)$,and $B$ and $C$ lie on the line $y=x+\alpha$ (where $\alpha$ is a variable),then the locus of the orthocenter of the triangle is:

The portion of the tangent to the curve $x^{2/3} + y^{2/3} = a^{2/3}, a > 0$ at any point, intercepted between the axes, is:

If $A=(5,3)$,$B=(3,-2)$ and a point $P$ is such that the area of the triangle $PAB$ is $9$,then the locus of $P$ represents

Ten ants are on the real line. At time $t=0$,the $k$-th ant starts at the point $k^2$ and,travelling at a uniform speed,reaches the point $(11-k)^2$ at time $t=1$. The number of distinct times at which at least two ants are at the same location is

$A$ straight line cuts off intercepts on the coordinate axes,the reciprocal of whose sum is $1/p$. Through which fixed point does this line always pass?

$A$ man standing on a railway platform noticed that a train took $21\,s$ to cross the platform (this means the time elapsed from the moment the engine enters the platform till the last compartment leaves the platform),which is $88\,m$ long,and that it took $9\,s$ to pass him. Assuming that the train was moving with uniform speed,what is the length of the train in meters?