We will learn how to form the equation of a circle when the centre of the circle coincides with the origin.
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 the circle coincides with the origin i.e., h = k = 0.
Then the equation (x  h)\(^{2}\) + (y  k)\(^{2}\) = a\(^{2}\) becomes x\(^{2}\) + y\(^{2}\) = a\(^{2}\)
Solved examples on the central form of the equation of a circle whose centre coincides with the origin:
1. Find the equation of the circle whose centre coincides with the origin and radius is √5 units.
Solution:
The equation of the circle whose centre coincides with the origin and radius is √5 units is x\(^{2}\) + y\(^{2}\) = (√5)\(^{2}\)
⇒ x\(^{2}\) + y\(^{2}\) = 5
⇒ x\(^{2}\) + y\(^{2}\)  5 = 0.
2. Find the equation of the circle whose centre coincides with the origin and radius is 10 units.
Solution:
The equation of the circle whose centre coincides with the origin and radius is 10 units is x\(^{2}\) + y\(^{2}\) = (10)\(^{2}\)
⇒ x\(^{2}\) + y\(^{2}\) = 100
⇒ x\(^{2}\) + y\(^{2}\)  100 = 0.
3. Find the equation of the circle whose centre coincides with the origin and radius is 2√3 units.
Solution:
The equation of the circle whose centre coincides with the origin and radius is 2√3 units is x\(^{2}\) + y\(^{2}\) = (2√3)\(^{2}\)
⇒ x\(^{2}\) + y\(^{2}\) = 12
⇒ x\(^{2}\) + y\(^{2}\)  12 = 0.
4. Find the equation of the circle whose centre coincides with the origin and radius is 13 units.
Solution:
The equation of the circle whose centre coincides with the origin and radius is 13 units is x\(^{2}\) + y\(^{2}\) = (13)\(^{2}\)
⇒ x\(^{2}\) + y\(^{2}\) = 169
⇒ x\(^{2}\) + y\(^{2}\)  169 = 0
5. Find the equation of the circle whose centre coincides with the origin and radius is 1 unit.
Solution:
The equation of the circle whose centre coincides with the origin and radius is 1 unit is x\(^{2}\) + y\(^{2}\) = (1)\(^{2}\)
⇒ x\(^{2}\) + y\(^{2}\) = 1
⇒ x\(^{2}\) + y\(^{2}\)  1 = 0
11 and 12 Grade Math
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