We will learn how to find the equation when the centre of a circle on xaxis.
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 centre of a circle is on the xaxis i.e., k = 0.
Then the equation (x  h)\(^{2}\) + (y  k)\(^{2}\) = a\(^{2}\) becomes (x  h)\(^{2}\) + y\(^{2}\) = a\(^{2}\) ⇒ x\(^{2}\) + y\(^{2}\)  2hx + h\(^{2}\) = a\(^{2}\) ⇒ x\(^{2}\) + y\(^{2}\)  2hx + h\(^{2}\) – a\(^{2}\) = 0
If the centre of a circle be on the xaxis, then the y coordinate of the centre will be zero. Hence, the general form of the equation of the circle will be of the form x\(^{2}\) + y\(^{2}\) + 2gx + c = 0, where g and c are the constants.
Solved examples on the central form of the equation of a circle whose centre is on the xaxis:
1. Find the equation of a circle whose centre of a circle is on the xaxis at 5 and radius is 9 units.
Solution:
Radius of the circle = 9 units.
Since, centre of a circle be on the xaxis, then the y coordinate of the centre will be zero.
The required equation of the circle whose centre of a circle is on the xaxis at 5 and radius is 9 units is
(x + 5)\(^{2}\) + y\(^{2}\) = 9\(^{2}\)
⇒ x\(^{2}\) + 10x + 25 + y\(^{2}\) = 81
⇒ x\(^{2}\) + y\(^{2}\) + 10x + 25  81 = 0
⇒ x\(^{2}\) + y\(^{2}\) + 10x  56 = 0
2. Find the equation of a circle whose centre of a circle is on the xaxis at 2 and radius is 3 units.
Solution:
Radius of the circle = 3 units.
Since, centre of a circle be on the xaxis, then the y coordinate of the centre will be zero.
The required equation of the circle whose centre of a circle is on the xaxis at 2 and radius is 3 units is
(x  2)\(^{2}\) + y\(^{2}\) = 3\(^{2}\)
⇒ x\(^{2}\)  4x + 4 + y\(^{2}\) = 9
⇒ x\(^{2}\) + y\(^{2}\)  4x + 4  9 = 0
⇒ x\(^{2}\) + y\(^{2}\)  4x  5 = 0
`11 and 12 Grade Math
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