Expressing a circle in a standard form expression is defined standard equation of a circle.

Imagining a circle in a plane at a particular distance from both axis of the Cartesian coordinate system is the standard form of the circle. Mathematically, a circle can be written in the form of a mathematical expression and it is actually possible by studying the relation of the circle with the Cartesian coordinate system.

Imagine a circle in the Cartesian coordinate system and assume the radius of the circle is

$r$units. Assume

$P$is the centre of the circle and it is located at

$a$units distance in horizontal

$x$-axis direction and

$b$units in vertical

$y$-axis direction from the origin. Therefore, the location of the point

$P$in the Cartesian coordinate system is

$P(a,b)$.

Consider a point on the circle and assume it represents all the points on the circle. It is assumed to call point

$Q$and the coordinates of the

$Q$in horizontal and vertical axis direction are

$x$and

$y$respectively. Therefore, the coordinates of the point

$Q$is

$Q(x,y)$in the Cartesian coordinate system.

Draw a line from point

$P$, and it must be parallel to the horizontal axis and draw another line from point

$Q$, and it should be perpendicular to the same axis and assume they both get intersected each other at a point and it is assumed to call point

$T$. Thus, a right angled triangle

$\Delta QPT$is formed inside the circle.

$QT,PT$

and

$PQ$are opposite side, adjacent side and hypotenuse of the right angled triangle

$\Delta QPT$.

The length of the opposite side of the right angled triangle

$\Delta QPT$is

$QT=OQ\u2013OT=y\u2013b$The length of the adjacent side of the right angled triangle

$\Delta QPT$is

$QT=OT\u2013OP=x\u2013a$The length of the hypotenuse of the right angled triangle

$\Delta QPT$is

$PQ=r$According to Pythagorean theorem, the relation between three sides can be expressed in a mathematical form as given here.

${PQ}^{2}={QT}^{2}+{PT}^{2}$

Substitute lengths of the three sides in this relation to get the equation of a circle in algebraic form expression.

${r}^{2}={(y\u2013b)}^{2}+{(x\u2013a)}^{2}$

It can be written as follows.

${(x\u2013a)}^{2}+{(y\u2013b)}^{2}={r}^{2}$

It is an algebraic expression which represents equation of a circle in standard form.

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