Find the equation of a line passing through the point $(4, 3)$,which cuts a triangle of minimum area from the first quadrant.

The locus of points $(x, y)$ in the plane satisfying $\sin ^2 x + \sin ^2 y = 1$ consists of

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:

Let $A=(4,0)$ and $B=(0,12)$ be two points in the plane. The locus of a point $C(x, y)$ such that the area of $\triangle ABC$ is $18$ sq units is

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

The Cartesian form of the polar equation $\theta = \tan^{-1} 2$ is