We will learn how to find the equation of a circle touches both xaxis and yaxis.
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 touches both xaxis and yaxis i.e., h = k = a.
Then the equation (x  h)\(^{2}\) + (y  k)\(^{2}\) = a\(^{2}\) becomes (x  a)\(^{2}\) + (y  a)\(^{2}\) = a\(^{2}\)
If a circle touches both the coordinate axes then the abscissa as well as ordinate of the centre will be equal to the radius of the circle. Hence, the equation of the circle will be of the form:
(x  a)\(^{2}\) + (y  a)\(^{2}\) = a\(^{2}\)
⇒ x\(^{2}\) + y\(^{2}\)  2ax  2ay + a\(^{2}\) = 0
Solved example on
the central form of the equation of a circle touches both xaxis and yaxis:
1. Find the equation of a circle whose radius is 4 units and touches both xaxis and yaxis.
Solution:
Radius of the circle = 4 units.
Since, the circle touches both xaxis and yaxis the centre of the circle is (4, 4).
The required equation of the circle whose radius is 4 units and touches both xaxis and yaxis is
(x  4)\(^{2}\) + (y  4)\(^{2}\) = 4\(^{2}\)
⇒ x\(^{2}\)  8x + 16 + y\(^{2}\)  8y + 16 = 16
⇒ x\(^{2}\)  8x  8y + 16 = 0
2. Find the equation of a circle whose radius is 8 units and touches both xaxis and yaxis.
Solution:
Radius of the circle = 8 units.
Since, the circle touches both xaxis and yaxis the centre of the circle is (8, 8).
The required equation of the circle whose radius is 8 units and touches both xaxis and yaxis is
(x  8)\(^{2}\) + (y  8)\(^{2}\) = 8\(^{2}\)
⇒ x\(^{2}\)  16x + 64 + y\(^{2}\)  16y + 64 = 64
⇒ x\(^{2}\) + y\(^{2}\)  16x  16y + 64 = 0
● The Circle
11 and 12 Grade Math
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