Question

The number of circles touching the line $y-x=0$ and the $y$ axis is
A. Zero
B. One
C. Two
D. Infinite

Answer 1

Hint: The group of points whose separation from a fixed point has a constant value are represented by a circle. The radius of the circle, abbreviated $r$, is a constant that describes this fixed point, which is known as the circle's centre. The formula for a circle$\left(\mathrm{x}-\mathrm{x}_{1}\right)^{2}+\left(\mathrm{y}-\mathrm{y}_{1}\right)^{2}=\mathrm{r}^{2}$ whose centre is at $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$

Complete step by step solution:

The location of a circle in the Cartesian plane is represented by a circle equation. We can write the equation for a circle if we know the location of the circle's center and how long its radius is. All of the points on the circle's circumference are represented by the circle equation.
The line $y-x=0$ and the $y$-axis are both touched by an unlimited number of circles.
Then there are infinite number of circles

Option ‘D’ is correct

Additional Information:
The radius of a circle is the length of the straight line that connects the center to any point on its circumference. Because a circle's circumference can contain an endless number of points, a circle can have more than one radius. This indicates that a circle has an endless number of radii and that each radius is equally spaced from the circle's center. When the radius's length varies, the circle's size also changes.

Note: If the radius is r, the center's coordinates are $(h,k),$and any point on the circle is$(x, y)$, the center of the circle formula is as follows:
$(x-h)^{2}+(y-k)^{2}=r^{2}$