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Solutions-on-Spheres_test

Q1) What is the distance from the point (2, 8, 10) to the xz-plane?

Sol) Distance =8

Q2) Find an equation of the sphere with center (2, -3, 4) and radius 5 ?

Sol) \((x-2)^{2}+(y+3)^{2}+(z-4)^{2}=5^{2}\)

      = \(x^{2}+y^{2}+z^{2}-4x+6y-8z+4=0\)

Q3) Find the center and radius of the sphere

        \(x^{2}+6x+y^{2}-4y+z^{2}-16z=-13 \)?

Sol) \((x+3)^{2}+(y-2)^{2}+(z-8)^{2}=3^{2}+2^{2}+8^{2}-13\)

       =    \((x+3)^{2}+(y-2)^{2}+(z-8)^{2}=64\)

Q4)  Find an equation of the largest sphere with center (5, 6 , 7) that is

        contained completely in the first octant ?

Sol) \((x-5)^{2}+(y-6)^{2}+(z-7)^{2}=5^{2}\)

Note: radius to choose min{5,6,7}=5

Q5) Find the equation of a sphere if one of its diameters has endpoints

      (-14, -12, -10) and (10, 12, 14) ?

Sol) Equation of Sphere (x+14)(x-10)+(y+12)(y-12)+(z+10)(z-14)=0

      \(x^{2}+y^{2}+z^{2}+4x-4z=140+144+140\)

    = \(x^{2}+y^{2}+z^{2}+4x-4z=424\)

Q6) Two spheres with the same radius r, one centered at (1,2,-4) and the other 

       centered at (2,4,3), touch each other tangentially at a single point. Find the

       value of the radius r.

Sol) 2r=\(\sqrt{(2-1)^{2}+(4-2)^{2}+(3+4)^{2}}\)

       2r=\(\sqrt{54}\)

        r=\(\frac{3\sqrt{6}}{2}\)




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