Let $A = (a, 0)$ and $B = (-a, 0)$ be two fixed points. For $a \in (-\infty, 0)$,point $P(x, y)$ moves in the plane such that $PA = nPB$ $(n \neq 0, 1)$. If $n = 1$,find the locus of point $P$.

$A$ rod of length $2l$ slides with its ends on two perpendicular lines. The locus of its mid-point is

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

There are two candles of the same length and size. Both of them burn at a uniform rate. The first one burns in $5 \, hr$ and the second one burns in $3 \, hr$. Both the candles are lit together. After how many minutes is the length of the first candle $3$ times that of the other?

If one vertex of an equilateral triangle of side $a$ lies at the origin and another vertex lies on the line $x - \sqrt{3}y = 0$,then the coordinates of the third vertex are:

The locus of the point of intersection of the straight lines $\frac{x}{a} + \frac{y}{b} = K$ and $\frac{x}{a} - \frac{y}{b} = \frac{1}{K}$, where $K$ is a non-zero real variable, is given by