$A$ ray of light passing through the point $A(1, 2)$ is reflected at a point $B$ on the $x$-axis and then passes through $(5, 3)$. Then the equation of $AB$ is:

$O$ is the origin and $A$ is the point $(3,4)$. If a point $P(x,y)$ moves such that the line segment $OP$ is always parallel to the line segment $OA$,then the equation of the locus of $P$ is:

The equation of the locus of a point which is equidistant from the points $(2, 3)$ and $(4, 5)$ is

If the point $(x, y)$ is equidistant from the points $(a + b, b - a)$ and $(a - b, a + b)$,then:

Let $A$ be the point $(0,4)$ and $B$ be a moving point on the $x$-axis. Let $M$ be the midpoint of $AB$ and let the perpendicular bisector of $AB$ meet the $y$-axis at $R$. The locus of the midpoint $P$ of $MR$ 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?