Find the Domain x^2+y^2-8x+14y+65=36

Problem

x2+y2−8*x+14*y+65=36

Solution

1. Group the terms containing x and the terms containing y together on one side of the equation.

x2−8*x+y2+14*y+65=36

2. Subtract the constant 65 from both sides to begin isolating the variable terms.

x2−8*x+y2+14*y=−29

3. Complete the square for the x terms by adding ((−8)/2)2=16 and for the y terms by adding (14/2)2=49 to both sides.

x2−8*x+16+y2+14*y+49=−29+16+49

4. Simplify the equation into the standard form of a circle (x−h)2+(y−k)2=r2

(x−4)2+(y+7)2=36

5. Identify the center (h,k) and the radius r from the standard form.

(h,k)=(4,−7)

r=√(,36)=6

6. Determine the domain by finding the interval of xvalues covered by the circle, which ranges from h−r to h+r

(x_min())=4−6=−2

(x_max())=4+6=10

Final Answer

Domain:[−2,10]

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