Math Doubts

Proof of Equation of a circle touches both axes

equation of circle touching both axes

The equation of a circle touches both axes, is written in the following three forms in mathematics.

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

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

$(3).\,\,$ $(x-r)^2+(y-r)^2$ $\,=\,$ $r^2$

You can use represent the circle touching the both axes by any one of the above three equations. So, let us learn how prove the equation of a circle touches both axes of two dimensional cartesian coordinate system in mathematical form geometrically.

Construction of a Triangle inside a Circle

A right triangle must be constructed inside a circle when the circle is touching both $x$-axis and $y$-axis in a quadrant of the two dimensional cartesian coordinate system and it is useful to derive the mathematical equation of a circle, which touches both axes.

geometric construction of right triangle for deriving equation of circle touching both axes

There are five geometrical steps to construct a right angled triangle inside a circle when the circle touches both $x$-axis and $y$-axis of the two dimensional space.

  1. Draw a circle in such away that it touches the $x$-axis at point $P$ and also touches the $y$-axis at point $Q$ in a two dimensional space.
  2. Denote the center or centre of a circle by $C$. Let’s assume that the $x$ and $y$ coordinates of the center are $a$ and $b$ respectively. Hence, the centre is written as $C(a, b)$ in coordinate form in mathematics.
  3. Consider a point on the circle and it is denoted by $R$. The coordinates of the point $R$ are denoted by $x$ and $y$ respectively. Therefore, the point $R$ is mathematically written as $R(x, y)$ in coordinate form.
  4. Connect the center $C$ and point $R$ by drawing a straight line and the line segment $\overline{CR}$ represents the radius of the circle. Let’s assume that the radius of the circle is $r$ units.
  5. Now, draw a horizontal line from the center to intersect the circle at a point. Similarly, draw a vertical line from point $R$ to intersect the horizontal line perpendicularly at a point $U$. It forms a right angled triangle, written as $\Delta UCR$ geometrically.

Relation of Center’s coordinates with Radius

It is time to understand the geometrical relationship between the radius and the coordinates of the circle.

geometric relation between the radius and coordinates of center or centre of circle

The distance from center to point $Q$ is the radius of the circle and it is exactly equal to the distance from origin to point $P$. Actually, it is the $y$-coordinate of the centre of the circle.

$\implies$ $CQ \,=\, OP$

$\,\,\,\therefore\,\,\,\,\,\,$ $r \,=\, b$

Similarly, the distance from point $P$ to centre is the radius of the circle and it is exactly equal to the distance from the origin to the point $Q$. In fact, it is the $x$-coordinate of the center of the circle.

$\implies$ $OQ \,=\, PC$

$\,\,\,\therefore\,\,\,\,\,\,$ $a \,=\, r$

The above two mathematical equations are cleared that the $x$-coordinate and $y$-coordinate are exactly equal to the radius of the circle.

$\,\,\,\therefore\,\,\,\,\,\,$ $a$ $\,=\,$ $b$ $\,=\,$ $r$

Measure the Lengths of sides of Right triangle

In $\Delta UCR$, the opposite side is $\overline{RU}$, the adjacent side is $\overline{CU}$ and the hypotenuse is $\overline{CR}$, and their lengths are written as $RU, CU$ and $CR$ respectively.

lengths of sides of right triangle inside circle touching both axes

Now, let us find the length of every side of the right angled triangle mathematically.

  1. $CU$ $\,=\,$ $QT$ $\,=\,$ $OT\,–\,OQ$ $\,=\,$ $x\,–\,a$
  2. $RU$ $\,=\,$ $SP$ $\,=\,$ $OS\,–\,OP$ $\,=\,$ $y-b$
  3. $CR \,=\, r$

Express the Relationship between the sides

According to the Pythagorean Theorem, the lengths of all three sides of right triangle $UCR$ can be written mathematically in an equation.

${CR}^2$ $\,=\,$ ${CU}^2+{RU}^2$

pythagorean theorem to lengths of sides of right triangle inside circle touching both axes

In this equation, $CU, CR$ and $RU$ represent the lengths of adjacent side, hypotenuse and opposite side. Now, substitute their values in the above mathematical equation.

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

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

It is a mathematical equation of a circle when the $C(a, b)$ is the center (or centre) and the radius of the circle is $r$ units. It is actually a standard equation of a circle.

We have derived that $a$ $\,=\,$ $b$ $\,=\,$ $r$, when the circle is touching the both axes. Now, the standard equation of a circle can be converted as an equation that represents a circle touches the both horizontal and vertical axes by this condition. Therefore, the equation of a circle touching both axes can be written in any one of the following forms in mathematics.

$(1).\,\,$ $(x-a)^2+(y-a)^2$ $\,=\,$ $a^2$ when $b$ and $r$ are replaced by $a$

$(2).\,\,$ $(x-b)^2+(y-b)^2$ $\,=\,$ $b^2$ when $a$ and $r$ are replaced by $b$

$(3).\,\,$ $(x-r)^2+(y-r)^2$ $\,=\,$ $r^2$ when $a$ and $b$ are replaced by $r$

Expansion

Take any one of the about three equations to express it in expansion form.

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

Now, expand each term on the left hand side of the equation as per the square of difference formula.

$\implies$ $x^2$ $+$ $a^2$ $-$ $2ax$ $+$ $y^2$ $+$ $a^2$ $-$ $2ay$ $\,=\,$ $a^2$

$\implies$ $x^2$ $+$ $y^2$ $-$ $2ax$ $-$ $2ay$ $+$ $a^2$ $+$ $a^2$ $\,=\,$ $a^2$

$\implies$ $x^2$ $+$ $y^2$ $-$ $2ax$ $-$ $2ay$ $+$ $a^2$ $+$ $a^2$ $-$ $a^2$ $\,=\,$ $0$

$\implies$ $x^2$ $+$ $y^2$ $-$ $2ax$ $-$ $2ay$ $+$ $a^2$ $+$ $\cancel{a^2}$ $-$ $\cancel{a^2}$ $\,=\,$ $0$

$\implies$ $x^2$ $+$ $y^2$ $-$ $2ax$ $-$ $2ay$ $+$ $a^2$ $\,=\,$ $0$

$\,\,\,\therefore\,\,\,\,\,\,$ $x^2$ $+$ $y^2$ $-$ $2a(x+y)$ $+$ $a^2$ $\,=\,$ $0$

It is an expansion of the equation of a circle when the circle is touching the both axes. Therefore, the above three forms can be written in expansion form as follows.

$(1).\,\,$ $x^2$ $+$ $y^2$ $-$ $2a(x+y)$ $+$ $a^2$ $\,=\,$ $0$

$(2).\,\,$ $x^2$ $+$ $y^2$ $-$ $2b(x+y)$ $+$ $b^2$ $\,=\,$ $0$

$(3).\,\,$ $x^2$ $+$ $y^2$ $-$ $2r(x+y)$ $+$ $r^2$ $\,=\,$ $0$

Other form

We have taken $C(a, b)$ is the center of a circle in coordinate form. In some countries, the centre of a circle in coordinate form is taken as $C(h, k)$, then the equation of a circle touches both axes is written as follows.

$(1).\,\,$ $(x-h)^2+(y-h)^2$ $\,=\,$ $h^2$

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

$(3).\,\,$ $(x-r)^2+(y-r)^2$ $\,=\,$ $r^2$

The above equations are expanded respectively in the following forms.

$(1).\,\,$ $x^2$ $+$ $y^2$ $-$ $2h(x+y)$ $+$ $h^2$ $\,=\,$ $0$

$(2).\,\,$ $x^2$ $+$ $y^2$ $-$ $2k(x+y)$ $+$ $k^2$ $\,=\,$ $0$

$(3).\,\,$ $x^2$ $+$ $y^2$ $-$ $2r(x+y)$ $+$ $r^2$ $\,=\,$ $0$

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