Show that the equation represents a sphere, and find its center and radius. x2+y2+z2+8x−6y+2z+17=0

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Show that the equation represents a sphere, and find its center and radius.  x2+y2+z2+8x−6y+2z+17=0

Mayme · Accepted Answer

Consider a sphere with center C(h, k, l) and radius r.  Formula:  Write the expression to find an equation of a sphere with center C (h, k, l) and radius r.  (x−h)2+(y−k)2+(z−l)2=r2(1)  Here,  (h, k, l) is the center of a sphere and  r is the radius of a sphere.  Rearrange the expression x2+y2+z2+8x−6y+2z+17=0 as follows.  (x2+8x+42−42)+(y2−6y+32−32)+(z2+2z+12−12)+17=0  (x2+8x+42)+(y2−6y+32)+(z2+2z+120+(−16−9−1)+17=0  (x+4)2+(y−3)2+(z+1)2=26−17  (x+4)2+(y−3)2+(z+1)2=9  [x−(−4)]2+(y−3)2+[z−(−1)]2=(3)2(2)  Equation (2) is similar to equation (1).  Therefore, the equation x62+y2+z2+8x−6y+2z+17=0 represents a sphere.  Compare equation (2) with equation (1).  h=−4  k=3  l=−1  r=3  Thus, the center of the spere is (-4, 3, -1) and the radius of the sphere is 3.

nick1337 · Accepted Answer

Step 1: Starting with the equation:x2+y2+z2+8x−6y+2z+17=0Let&#039;s group the terms involving x, y, and z respectively:(x2+8x)+(y2−6y)+(z2+2z)+17=0To complete the square for x, we add the square of half the coefficient of x to both sides of the equation:(x2+8x+16)+(y2−6y)+(z2+2z)+17=16Simplifying the expression:(x+4)2+(y2−6y)+(z2+2z)+17=16Step 2: Next, let&#039;s complete the square for y:(x+4)2+(y2−6y+9)+(z2+2z)+17=16+9Simplifying further:(x+4)2+(y−3)2+(z2+2z)+17=25Step 3: Finally, we complete the square for z:(x+4)2+(y−3)2+(z2+2z+1)+17=25+1Simplifying again:(x+4)2+(y−3)2+(z+1)2+17=26Now we have the equation in the standard form of a sphere:(x−(−4))2+(y−3)2+(z−(−1))2=26Thus, the center of the sphere is (−4,3,−1), and the radius is 26.

RizerMix · Accepted Answer

Answer:Center (−4,3,−1) and radius r=4Explanation:The given equation is: x2+y2+z2+8x−6y+2z+17=0To complete the square, we group the x terms, y terms, and z terms separately:(x2+8x)+(y2−6y)+(z2+2z)+17=0Now, we add the necessary terms to complete the square for each variable:(x2+8x+16)+(y2−6y+9)+(z2+2z+1)+17=16+9+1Simplifying the equation:(x+4)2+(y−3)2+(z+1)2=16Comparing this with the standard form of a sphere equation:(x−h)2+(y−k)2+(z−l)2=r2We can see that the given equation represents a sphere with center (−4,3,−1) and radius r=4.

Vasquez · Accepted Answer

Let&#039;s start by completing the square for each variable in the given equation:(x2+8x)+(y2−6y)+(z2+2z)+17=0To complete the square for x, we add (82)2=16 to both sides of the equation:(x2+8x+16)+(y2−6y)+(z2+2z)+17=16Similarly, to complete the square for y and z, we add (−62)2=9 and (22)2=1 respectively:(x2+8x+16)+(y2−6y+9)+(z2+2z+1)+17=16+9+1Now, we can simplify the equation:(x+4)2+(y−3)2+(z+1)2=26Comparing this equation with the standard form, we can identify the center and radius of the sphere. The center is the opposite of the coefficients inside the parentheses, so the center of the sphere is (−4,3,−1). The radius can be determined by taking the square root of the constant term on the right side, so the radius is 26.Therefore, the equation x2+y2+z2+8x−6y+2z+17=0 represents a sphere with center (−4,3,−1) and radius 26.

Show that the equation represents a sphere, and find its center and radius. x^{2} + y^{2} + z^{2} + 8x - 6y + 2z + 17 =0

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