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Question

How do you find the equation of a circle center of (5,-2) and a radius of 2?


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Solution

Step-1: Discuss the equation of a circle

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

The general equation of a circle is given by (x-α)2+(y-β)2=r2---(1) a center of (0,0) and a radius of r.

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.

Step-2: Determine the equation of the circle:

Put the values of the center and the radius in the equation (1) :

Evaluate the expression :

(x(5))2+(y(2))2=22(x5)2+(y+2)2=22

Solve the equation using the following identities :

(x+y)2=x2+y2+2xy(x-y)2=x2+y2-2xy

Evaluate the expression further :

x2+5225x+y2+22+22y=4x2+2510x+y2+4+4y=4x2+y210x+4y=4254x2+y210x+4y=25x2+y210x+4y+25=0

Hence, a circle having its center as (5,-2) and radius =2 units will have its equation as

x2+y2-10x+4y+25=0


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