Question

How to find the standard form of the equation of the specified circle given by Centre: $(0,0)$ and radius: $9$?

Answer 1

Hint: In this question, they have given the centre and the radius and asked to find the standard form of the equation of the specified circle. We need to use the below mentioned formula and represent the given values in it to get the required answer.

Formulas used: Standard Equation of a circle:
\[{(x - a)^2} + {(y - b)^2} = {r^2}\], where $(a,b)$ are the coordinates of the centre and $r$ is radius

Complete step by step answer:
In this question, they have given the centre and the radius and asked to find the standard form of the equation of the specified circle.
The given centre: $(0,0)$
The given radius: $9$
We use the formula to form an equation of the specified circle
\[{(x - a)^2} + {(y - b)^2} = {r^2}\],where $(a,b)$ are the coordinates of the centre and $r$ is radius.
Clearly, we have a given data as follows: $(a,b)$= $(0,0)$and $r = 9$
Simply substituting the given values in the above mentioned formula will form the standard equation of the circle specified.
Therefore the standard form of equation of a circle in the formula and we get
\[ \Rightarrow {(x - 0)^2} + {(y - 0)^2} = {9^2}\]
On simplify we get
\[ \Rightarrow {x^2} + {y^2} = 81\]
therefore, \[{x^2} + {y^2} = 81\] is the standard form of the equation of the given specified circle.
Hence, we got the required answer.

Note: The equation of a circle is a rule satisfied by the coordinates $(x,y)$ of any point that lies on the circumference. Points that do not lie on the circle will not satisfy the equation. The equation of a circle will vary depending on its size (radius) and its position on the Cartesian plane.
This equation of circle is derived from the great Pythagoras theorem.