Question: If the abscissa and the ordinate of a point in a locus increase at the same rate then, the locus of the point represents
a.) family of concentric circles
b.) family of parabola with same vertex
c.) family of parallel lines with slope=1
d.) an empty set
I am looking forward to an explanation of each of the given options and then the solution to the question.
Let $P(x,y)$ be a point in a locus such that $x$= abscissa of $P$ and $y$= ordinate of $P$. Since, $x$ and $y$ increase at the same rate, we can write
$\frac{dy}{dt}$ = $\frac{dx}{dt}$
Integrating on both sides, we get,
$y=x+c$, a straight line with slope(m) = $1$ and y-intercept= $c$
Now, we get to the conclusion that the function which represents the locus(where point P lies) is a straight line. So, the options a.) and b.) are eliminated.
Option d.) is also eliminated since we have obtained a straight line, so it's not null.
The solution would be option c.) as $y=x+c$ is a family of parallel lines with slope(m)=$1$ for $(x,y) \in R$.