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

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

Alternate coordinate systems
asked 2020-12-25
Show that the equation represents a sphere, and find its center and radius.
\(x^{2} + y^{2} + z^{2} + 8x - 6y + 2z + 17 =0\)

Answers (1)

Consider a sphere with center C(h, k, l) and radius r.
Write the expression to find an equation of a sphere with center C (h, k, l) and radius r.
\((x - h)^{2} + (y - k)^{2} + (z - l)^{2} = r^{2} (1)\)
(h, k, l) is the center of a sphere and
r is the radius of a sphere.
Rearrange the expression \(x^{2} + y^{2} + z^{2} + 8x - 6y + 2z + 17 = 0\) as follows.
\((x^{2} + 8x + 4^{2} - 4^{2}) + (y^{2} - 6y + 3^{2} - 3^{2}) + (z^{2} + 2z + 1^{2} - 1^{2}) + 17 = 0\)
\((x^{2} + 8x +4^{2}) + (y^{2} - 6y + 3^{2}) + (z^{2} + 2z + 1^{20} + (-16 - 9 - 1) + 17 = 0\)
\((x + 4)^{2} + (y - 3)^{2} + (z + 1)^{2} = 26 - 17\)
\((x + 4)^{2} + (y - 3)^{2} + (z + 1)^{2} = 9\)
\([x - (-4)]^{2} + (y - 3)^{2} + [z - (-1)]^{2} = (3)^{2} (2)\)
Equation (2) is similar to equation (1).
Therefore, the equation \(x62 + y^{2} + z^{2} + 8x - 6y + 2z + 17 = 0\) 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.

Relevant Questions

asked 2021-02-18
Two hollow metal spheres are concentric with each other. Theinner sphere has a radius of 0.131 m and a potential of 89.6 V. Theradius of the outer sphere is 0.155 m and its potential is 78.3 V.If the region between the spheres is filled with Teflon, find theelectric energy contained in this space.
asked 2021-05-22
Find the equation of the tangent plane to the graph of \(\displaystyle{f{{\left({x},{y}\right)}}}={8}{x}^{{{2}}}-{2}{x}{y}^{{{2}}}\) at the point (5,4).
asked 2021-04-25
The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius \(\displaystyle{R}={7.4}\times{10}^{{-{15}}}\) m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
A. Find the radii of the two "daughter" nuclei of charge+46e.
B. In a simple model for the fission process, immediatelyafter the uranium-236 nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit \(\displaystyle={1.66}\times{10}^{{-{27}}}\) kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes).
asked 2021-03-05
Consider the solid that is bounded below by the cone \(z = \sqrt{3x^{2}+3y^{2}}\)
and above by the sphere \(x^{2} +y^{2} + z^{2} = 16.\).Set up only the appropriate triple integrals in cylindrical and spherical coordinates needed to find the volume of the solid.
asked 2021-05-08
A high-speed sander has a disk 4.00 cm in radius that rotates about its axis at aconstant rate of 1265 rev/min.Determine
(a) the angular speed of the disk in radians persecond,
(b) the linear speed of a point 2.2 cmfrom the disk's center,
(c) the centripetal acceleration of a point on the rim, and
(d) the total distance traveled by a point on the rim in1.96 s.
asked 2021-05-19
Forces F1=7.50N and F2=5.30N are applied tangentially to a wheel with radius 0.330m. What is the net trque on the wheel due tothese two forces for an axis perpendicular to the wheel and passingthrough its center.
asked 2021-02-19
A uniform 8.40kg, spherical shell 50.0cm in diameter has foursmall 2.00kg masses attached to its outer surface and equallyspaced around it. This combination is spinning about an axisrunning through the center of the sphere and two of the smallmasses. What friction torque is needed to reduce its angular speedfrom 75.0rpm to 50.0rpm in 30.0s?
asked 2020-11-02
Find inequalities that describe the solid and state the coordinate system used. Position the solid on the coordinate system such that the inequalities are as simple as possible. "The solid that remains after a hole 1 inch in diameter is drilled through the center of a sphere 6 inches in diameter "
asked 2021-05-20
Assume that a ball of charged particles has a uniformly distributednegative charge density except for a narrow radial tunnel throughits center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton any where along the tunnel or outside the ball. Let \(\displaystyle{F}_{{R}}\) be the magnitude of the electrostatic force on the proton when it islocated at the ball's surface, at radius R. As a multiple ofR, how far from the surface is there a point where the forcemagnitude is 0.44FR if we move the proton(a) away from the ball and (b) into the tunnel?
asked 2021-03-03
Figure shows a nonconducting rod with a uniformly distributed charge +Q. The rod forms a 10/22 of circle with radius R and produces an electric field of magnitude Earc at its center of curvature P. If the arc is collapsed to a point at distance R from P, by what factor is the magnitude of the electric field at P multiplied?