Question

The locus of the center of the circles which touches both the axes is given by
A)$\quad x^{2}-y^{2}=0$
B)$\quad x^{2}+y^{2}=0$
C)$\quad x^{2}-y^{2}=1$
D)$\quad x^{2}+y^{2}=1$

Answer 1

Hint: A locus is a group of locations whose positions are indicated by particular circumstances. For instance, various independence movements have originated in a stretch of the southwest. The locus is referred to in this context as any location's center.

Complete step by step Solution:
The center of a circle is a location inside the circle that is situated in the middle of the circumference. The radius of a circle is the constant distance from the circle's center to any point on the circle.A circle's diameter is defined as the segment of a line that connects two locations on the circle and passes through its center.
Since, the circle touches both the axes,
then $g=f$.
$\Rightarrow \quad \mathrm{g}^{2}-\mathrm{f}^{2}=0$
$\Rightarrow \quad \mathrm{x}^{2}-\mathrm{y}^{2}=0$
Which is the required locus of the centers of the circles which touch both the axes.

Hence, the correct option is (A).

Note: A locus is a group of points with positions that meet the requirements for their locations and create geometric structures like lines, lines segments, circles, curves, etc. Instead of just being a collection of points, the points can also be thought of as locations or points that can be moved.
The circle, in terms of the locus of points or loci, corresponds to all points that are equally spaced apart from one fixed point, with the fixed point being the circle's centre and its radius being the collection of points outward from the centre.