Question

How to find the standard form of the equation of the specified circle given centre: (0,0); Radius: 9?

Answer 1

Hint: According to the question given in the question we have to determine the standard equation of the circle having Centre: (0,0); Radius: 9. So, first of all to determine the standard form of the equation of the specified circle we have to understand about the standard form of the equation of the circle which is as explained below:
Standard form of the equation of the circle:
\[ \Rightarrow {(x - {x_1})^2} + {(y - {y_1})^2} = {r^2}...............(A)\]
Where, $({x_1},{y_1})$ are the centres of the circle and r is the radius of the standard circle.
Now, as we know that the centre of the circle is (0,0) and the radius of the circle is 9. So, we have to compare it with the standard form of the equation if the circle (A) as mentioned above so that we can substitute the centre and radius in the equation of the circle.
Now, we have to solve the expression as obtained after substituting all the terms which are centre and radius in the standard form of the equation of the circle.

Complete step-by-step solution:
Step 1: First of all to determine the standard form of the equation of the specified circle we have to understand about the standard form of the equation (A) of the circle which is as explained in the solution hint.
Step 2: Now, as we know that the centre of the circle is (0,0) and the radius of the circle is 9. So, we have to compare it with the standard form of the equation if the circle (A) as mentioned above so that we can substitute the centre and radius in the equation of the circle. Hence,
$ \Rightarrow $Centre: $({x_1},{y_1}) = (0,0)$
$ \Rightarrow $Radius: $r = 9$
Step 3: Now, we have to solve the expression as obtained after substituting all the terms which are centre and radius in the standard form of the equation of the circle. Hence,
$
   \Rightarrow {(x - 0)^2} + {(y - 0)^2} = {(9)^2} \\
   \Rightarrow {x^2} + {y^2} = 81
 $

Hence, with the help of the standard form of the equation (A) we have determined the standard form of the equation having centre (0,) and radius 9 is $ \Rightarrow {x^2} + {y^2} = 81$.

Note: To obtain the standard form of the equation of the circle it is necessary that we have to determine the centre and radius for that and if that is already given then we just have to substitute them.
If we have to find the centre and radius of the equation of the circle then we just have to compare the given equation of the circle with the standard form of the equation to obtain the radius and centre.