We will learn how to find the equation of a circle passes through the origin and centre lies on y-axis.

The equation of a circle with centre at (h, k) and radius equal to a, is (x - h)\(^{2}\) + (y - k)\(^{2}\) = a\(^{2}\).

When the circle passes through the origin and centre lies on x-axis i.e., h = 0 and k = a.

Then the equation (x - h)\(^{2}\) + (y - k)\(^{2}\) = a\(^{2}\) becomes x\(^{2}\) + (y - a)\(^{2}\) = a\(^{2}\)

If a circle passes through the origin and centre lies on y-axis then the y co-ordinate will be equal to the radius of the circle and the abscissa of the centre will be zero. Hence, the equation of the circle will be of the form:

x\(^{2}\) + (y - a)\(^{2}\) = a\(^{2}\)

⇒ x\(^{2}\) + y\(^{2}\) - 2ay = 0

Solved example on
the central form of the equation of a circle passes through the origin and
centre lies on y-axis:

**1.**
Find the equation of a circle
passes through the origin and centre lies on y-axis at (0, -6).

**Solution:**

Centre of the lies on x-axis at (0, -6)

Since, circle passes through the origin and centre lies on y-axis then the y co-ordinate will be equal to the radius of the circle and the abscissa of the centre will be zero.

The required equation of the circle passes through the origin and centre lies on y-axis at (0, -6) is

x\(^{2}\) + (y + 6)\(^{2}\) = (-6)\(^{2}\)

⇒ x\(^{2}\) + y\(^{2}\) + 12y + 36 = 36

⇒ x\(^{2}\) + y\(^{2}\) + 12y = 0

**2.**
Find the equation of a circle
passes through the origin and centre lies on y-axis at (0, 20).

**Solution:**

Centre of the lies on y-axis at (0, 20)

Since, circle passes through the origin and centre lies on y-axis then the y co-ordinate will be equal to the radius of the circle and the abscissa of the centre will be zero.

The required equation of the circle passes through the origin and centre lies on y-axis at (0, 20) is

x\(^{2}\) + (y - 20)\(^{2}\) = 20\(^{2}\)

⇒ x\(^{2}\) + y\(^{2}\) - 40y + 400 = 400

⇒ x\(^{2}\) + y\(^{2}\) - 40y = 0

**●** **The Circle**

**Definition of Circle****Equation of a Circle****General Form of the Equation of a Circle****General Equation of Second Degree Represents a Circle****Centre of the Circle Coincides with the Origin****Circle Passes through the Origin****Circle Touches x-axis****Circle Touches y-axis****Circle Touches both x-axis and y-axis****Centre of the Circle on x-axis****Centre of the Circle on y-axis****Circle Passes through the Origin and Centre Lies on x-axis****Circle Passes through the Origin and Centre Lies on y-axis****Equation of a Circle when Line Segment Joining Two Given Points is a Diameter****Equations of Concentric Circles****Circle Passing Through Three Given Points****Circle Through the Intersection of Two Circles****Equation of the Common Chord of Two Circles****Position of a Point with Respect to a Circle****Intercepts on the Axes made by a Circle****Circle Formulae****Problems on Circle**

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