Math Doubts

Equation of a circle when centre of the circle lies on y-axis

Equation

$x^2 + {(y-b)}^2 = r^2$
$x^2 + y^2 -2by + b^2 -r^2 = 0$

Proof

Consider a circle having radius of $r$ units in two dimensional Cartesian coordinate system. The centre of circle lies on $y$-axis without touching the $x$-axis. The centre of the circle is $b$ units distance from the origin. Therefore, the centre of circle is $C(0, b)$. Take $P$ as a point on the circumference of the circle and its coordinates are $x$ and $y$. Therefore, the point is $P(x, y)$.

centre of the circle lies on y axis

Draw a line from point $C$ and it is parallel to horizontal axis and also draw another straight line from point $P$ and it should be parallel to vertical axis. The two lines intersect at point $Q$. Thus, a right angled triangle $PCQ$ is constructed geometrically.

According to right angled triangle $PCQ$.

  1. Length of the opposite side is $PQ = OP \,-\, OQ = y-b$
  2. Length of the adjacent side is $CQ = x$
  3. Length of the hypotenuse $CP = r$

The lengths of all three sides are known geometrically. Express the relation between them in mathematical form by using Pythagorean theorem.

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

$\implies r^2 = x^2 + {(y-b)}^2$

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

It is an equation of the circle in general compact form when the centre of the circle lies on the $y$-axis without touching the $x$-axis.

Use expansion of square of difference of two terms to expand this general equation of circle.

$\implies x^2 + y^2 + b^2 -2by = r^2$

$\implies x^2 + y^2 -2by + b^2 = r^2$

$\therefore \,\,\,\,\, x^2 + y^2 -2by + b^2 -r^2 = 0$



Follow us
Email subscription
Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Mobile App for Android users Math Doubts Android App
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more