We will learn how to find the equation of a circle passes through the origin and centre lies on x-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 = a and k = 0.

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

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

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

⇒ x\(^{2}\) + y\(^{2}\) - 2ax = 0

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

**1.**
Find the equation of a circle
passes through the origin and centre lies on y-axis at (0, -2).

**Solution:**

Centre of the lies on y-axis at (0, -2)

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

The required equation of the circle passes through the origin and centre lies on y-axis at (0, 2) is

(x + 7)\(^{2}\) + y\(^{2}\) = (-7)\(^{2}\)

⇒ x\(^{2}\) + 14x + 49 + y\(^{2}\) = 49

⇒ x\(^{2}\) + y\(^{2}\) + 14x = 0

**2.**
Find the equation of a circle
passes through the origin and centre lies on x-axis at (12, 0).

**Solution:**

Centre of the lies on x-axis at (12, 0)

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

The required equation of the circle passes through the origin and centre lies on x-axis at (12, 0) is

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

⇒ x\(^{2}\) - 24x + 144 + y\(^{2}\) = 144

⇒ x\(^{2}\) + y\(^{2}\) - 24x = 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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