Two circles have equations of x2 + y2 – 4x – 4y + 4 = 0 and x2 + y2 – 4x + 8y + 4 = 0. Compute the length of the common external tangent.

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karton · Accepted Answer

To find the length of the common external tangent between the two circles, we can first determine their centers and radii.Let&#039;s start with the equation of the first circle:x2+y2−4x−4y+4=0To rewrite this equation in standard form, we complete the square for both x and y terms:(x2−4x)+(y2−4y)=−4(x2−4x+4)+(y2−4y+4)=−4+4+4(x−2)2+(y−2)2=4From this equation, we can see that the center of the first circle is (2,2) and the radius is 4=2.Now, let&#039;s move on to the equation of the second circle:x2+y2−4x+8y+4=0Completing the square for both x and y terms:(x2−4x)+(y2+8y)=−4(x2−4x+4)+(y2+8y+16)=−4+4+16(x−2)2+(y+4)2=16From this equation, we can determine that the center of the second circle is (2,−4) and the radius is 16=4.To find the length of the common external tangent, we need to find the distance between the centers of the two circles and then subtract the sum of their radii.The distance between the centers is given by the distance formula:d=(x2−x1)2+(y2−y1)2Substituting the values of the centers (2,2) and (2,−4) into the distance formula, we have:d=(2−2)2+(−4−2)2d=0+36=36=6The sum of the radii is 2+4=6.Finally, we can calculate the length of the common external tangent by subtracting the sum of the radii from the distance between the centers:Length of the common external tangent = d−(r1+r2)=6−6=0.Therefore, the length of the common external tangent between the two circles is 0.

Two circles have equations of x2 +

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Two circles have equations of x2 + y2 – 4x – 4y + 4 = 0 and x2 + y2 – 4x + 8y + 4 = 0. Compute the length of the common external tangent.

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