Question

How do you find the centre of the circle with equation \[{(x - 3)^2} + {(y + 4)^2} = 25\]?

Answer 1

Hint: A circle is a closed two-dimensional figure in which the set of all the points in the plane is equidistant from a given point called centre.
We know that the standard equation of a circle with centre \[(a,b)\] and radius \[r\] is
\[{(x - a)^2} + {(y - b)^2} = {r^2}\]
By comparing the given equation with the standard form of circle we get the centre of the circle.

Complete Step by Step Solution:
It is given that, the equation of the circle \[{(x - 3)^2} + {(y + 4)^2} = 25\]
We have to find the centre of the circle \[{(x - 3)^2} + {(y + 4)^2} = 25\].
We know that the standard equation of a circle with centre \[(a,b)\] and radius \[r\] is
\[{(x - a)^2} + {(y - b)^2} = {r^2}\]
Comparing the given equation with general equation of circle we get,
\[a = 3,b = - 4\]
So, the centre is \[(3, - 4)\].

Hence, the centre of the circle with equation \[{(x - 3)^2} + {(y + 4)^2} = 25\]is \[(3, - 4)\].

Note: In Maths or Geometry, a circle is a special kind of ellipse in which the eccentricity is zero and the two foci are coincident. A circle is also termed as the locus of the points drawn at an equidistant from the centre.
The distance from the centre of the circle to the outer line is its radius. Diameter is the line which divides the circle into two equal parts and is also equal to twice the radius.
Circle equation formula refers to the equation of a circle which represents the centre-radius form of the circle.
To recall, a circle is referred to a round shape boundary where all the points on the boundary are equidistant from the centre. An equation is generally required to represent the circle. There are basically two forms of representation: Standard Form and General Form.
The standard equation or form of the circle is \[{(x - a)^2} + {(y - b)^2} = {r^2}\], where, \[(a,b)\] is the centre and \[r\] is the radius.
The general equation or form of the circle is \[{x^2} + {y^2} + Ax + By + c = 0\], where, \[A,B,c\] are constants.