Question

Find the equation of the circle which passes through two points on the x-axis which is the distance of ${\text{4units }}$from the origin and whose radius is ${\text{5unit}}$.

Answer 1

Here there are two possible circles, one is with the maximum area lying above the x-axis and another is the maximum area lying below the x-axis. Now, drawing the circle by taking symmetry on the x-axis. And find the distance of the line perpendicular to the axis and passing through the center so the coordinate can be determined and as the radius is known we can calculate the equation of circle using the center and radius method.

Complete step by step Answer:

Diagram :

It is given that the points are at the distance of ${\text{4units }}$ from the origin and so the coordinates of A and B are
${\text{( - 4,0),(4,0)}}$
Now, as given that the radius is of ${\text{5unit}}$ and the distance from the origin to coordinate A and B is${\text{4units }}$and they are at the right angle so calculate the distance from the origin to the center of the circle using Pythagoras theorem.
So,

$$\sqrt {{\text{O}}{{\text{C}}^{\text{2}}}{\text{ + O}}{{\text{B}}^{\text{2}}}} {\text{ = BC}} \\\ \Rightarrow \sqrt {{\text{O}}{{\text{C}}^{\text{2}}}{\text{ + }}{{\text{4}}^{\text{2}}}} {\text{ = 5}} \\\ \Rightarrow {\text{O}}{{\text{C}}^{\text{2}}}{\text{ = 25 - 16 = 9}} \\\ \Rightarrow {\text{OC = 3unit}} \\\$$

And from here it is cleared that the two possible coordinates of both the circle’s centre are ${\text{(0,3),(0, - 3)}}$.
Now, as we know the centres of both the circle and their radius are also known so we write the equation of circle as ,

$${{\text{(x - 0)}}^{\text{2}}}{\text{ + (y $\pm$ 3}}{{\text{)}}^{\text{2}}}{\text{ = 25}} \\\ \Rightarrow {{\text{x}}^{\text{2}}}{\text{ + }}{{\text{y}}^{\text{2}}}{\text{ $\pm$ 6y + 9 = 25}} \\\ \Rightarrow {{\text{x}}^{\text{2}}}{\text{ + }}{{\text{y}}^{\text{2}}}{\text{ $\pm$ 6y = 16}} \\\$$

Hence, ${{\text{x}}^{\text{2}}}{\text{ + }}{{\text{y}}^{\text{2}}}{\text{ $\pm$ 6y = 16}}$ are our required equation of circle.

Note: A circle is a shape consisting of all points in a plane that are a given distance from a given point, the center; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant.