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# A circle has its center at (-2, 5) and a radius of 4 units. What is the equation of the circle?

(x + 2)^{2} + (y - 5)^{2} = 16

(x + 2)^{2} + (y + 5)^{2} = 16

(x + 2)^{2} + (y - 5)^{2} = 4

(x - 2)^{2} + (y + 5)^{2} = 4

**Solution: **

Given: Center of the circle = (-2 ,5) and radius = 4 units.

We know the general equation of a circle is given by (x - α)^{2} +(y - β)^{2} = r^{2} --- (1)

Where (x, y) is any point on the given circle, (α, β) is the center of the given circle and r is the radius of the given circle.

Now, put the values of the center and the radius in the equation (1)

We get,

⇒ (x - (-2))^{2} +(y - 5 )^{2} = 4^{2}

⇒ (x + 2)^{2} + (y - 5)^{2} = 4^{2}

Now solving the equation using the following identities

(x + y)^{2} = x^{2} + y^{2} + 2xy

(x - y)^{2} = x^{2} + y^{2} - 2xy

⇒ x^{2} + 2^{2} + 2.2.x + y^{2} + 5^{2} - 2.5.y = 4^{2}

⇒ x^{2} + 4 + 4x + y^{2} + 25 - 10y = 16

⇒ x^{2} + y^{2} + 4x - 10y = 16 - 25 - 4

⇒ x^{2} + y^{2} + 4x - 10y = -13

⇒ x^{2} + y^{2} + 4x - 10y + 13 = 0

## A circle has its center at (-2, 5) and a radius of 4 units. What is the equation of the circle?

**Summary: **

A circle having its center as (-2, 5) and radius = 4 units will have its equation as (x + 2)^{2} + (y - 5)^{2} = 4^{2}.

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