Math Doubts

Proof of standard equation of a circle

geometric proof of standard circle equation

The equation of a circle in standard form is popularly written in the following two forms in mathematics.

$(1).\,\,\,$ $(x-a)^2+(y-b)^2$ $\,=\,$ $r^2$

$(2).\,\,\,$ $(x-h)^2+(y-k)^2$ $\,=\,$ $r^2$

Now, let’s learn how to prove the standard form equation of a circle mathematically as per the geometric system when the circle does not touch both axes in two dimensional cartesian coordinate system.

Construction of a Triangle inside a Circle

A right triangle or right angled triangle should be constructed inside a circle when the circle does not touch both $x$ axis and $y$ axis in a quadrant of two-dimensional cartesian coordinate system. It helps us to express the equation of a circle in mathematical form.

construction of right angle triangle inside a circle

The following five steps are the steps for the geometrical construction of a right angled triangle inside a circle when the circle does not touch both axes.

  1. Consider the first quadrant of bi-dimensional cartesian coordinate system and draw a circle in such a way that it does not touch any axis.
  2. Let’s denote the center (or centre) of a circle by $C$. Assume that, the centre (or center) is $a$ and $b$ units away from the origin in horizontal and vertical directions respectively. So, the center (or centre) is expressed as $C(a, b)$ in coordinate form mathematically.
  3. $P$ is a point on the circumference of circle. Assume that the point $P$ is $x$ and $y$ units away from the origin in horizontal and vertical directions. Therefore, the point $P$ is written as $P(x, y)$ in coordinate form.
  4. Now, draw a straight line from center $C$ to point $P$ for connecting them and the line segment $\overline{CP}$ represents the radius in the circle. Assume that the length of the radius is $r$ units.
  5. Finally, draw a straight line horizontally from centre (or center) until it touches a point on the circumference of circle. Similarly, draw a straight line from $P$ in downward direction until it intersects the horizontal line at point $Q$. Thus, right angled triangle is formed geometrically and it is written as $\Delta QCP$ in mathematics.

Find the lengths of all sides of Triangle

In $\Delta QCP$, $\overline{PQ}$ is opposite side, $\overline{CQ}$ is adjacent side and $\overline{CP}$ is hypotenuse, and the lengths of them are written as $PQ, CQ$ and $CP$ respectively.

lengths of right angle triangle inside a circle

Now, let’s calculate the length of every side in the right triangle.

  1. $CQ$ $\,=\,$ $OQ\,–\,OC$ $\,=\,$ $x \,–\,a$
  2. $PQ$ $\,=\,$ $OP\,–\,OQ$ $\,=\,$ $y\,-\,b$
  3. $CP \,=\, r$.

Write the relation between the sides

The length of each side of right triangle $QCP$ is evaluated and it is time express the mathematical relation between the lengths of the sides. It can be done as per the Pythagorean Theorem.

pythagorean theorem to right angle triangle inside a circle

${CP}^2$ $\,=\,$ ${CQ}^2+{PQ}^2$

Here, $CP, CQ$ and $PQ$ are the lengths of the hypotenuse, adjacent side and opposite side respectively. Now, substitute the lengths of the sides in the above equation.

$\implies$ $r^2$ $\,=\,$ $(x-a)^2+(y-b)^2$

$\,\,\,\therefore\,\,\,\,\,\,$ $(x-a)^2+(y-b)^2$ $\,=\,$ $r^2$

It is called the equation of a circle in standard form when the $C(a, b)$ is center (or centre) in coordinate form and the radius of the circle is $r$ units.

Other form

The equation of a circle in standard form is also popularly written in the following form when the coordinates of center (or centre) are denoted by $h$ and $k$. It the centre (or center) in coordinate form is written as $C(h, k)$ and the radius is denoted by $r$.

$(x-h)^2+(y-k)^2$ $\,=\,$ $r^2$