Question

The equation of the circle whose diameters have the end points $(a,0)$ and $(0,b)$

Answer 1

Hint: A circle's diameter is determined by multiplying the radius by two. The diameter is measured from one end of the circle to a point on the other end, passing through the center, whereas the radius is measured from the center of a circle to one endpoint on the circle's perimeter. The letter D is used to identify it. A circle has an infinite number of points on its circumference, which translates to an endless number of diameters with equal lengths for each diameter.

Complete step by step solution:
Let O be the center
The center of the circle will be
${O} \equiv\left[\dfrac{0+{a}}{2}, \dfrac{0+{b}}{2}\right] $
${O} \equiv\left(\dfrac{{a}}{2}, \dfrac{{b}}{2}\right)$
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.
Radius $=\sqrt{\left({a}-\dfrac{{a}}{2}\right)^{2}+\left(\dfrac{b}{2}\right)^{2}}$
$=\sqrt{\dfrac{{a}^{2}}{4}+\dfrac{b^{2}}{4}}=\sqrt{\dfrac{{a}^{2}+b^{2}}{4}}=\dfrac{\sqrt{a^{2}+b^{2}}}{2}$
Then the equation of the circle is
${{\left( \text{x}-\dfrac{\text{a}}{2} \right)}^{2}}+{{\left( \text{y}-\dfrac{\text{b}}{2} \right)}^{2}}=\dfrac{{{\text{a}}^{2}}+{{b}^{2}}}{4}$

Note: 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.