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MATHMAA

# 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) $$(x-5)^{2}+(y-6)^{2}+(z-7)^{2}=5^{2}$$

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

Sol) 2r=$$\sqrt{(2-1)^{2}+(4-2)^{2}+(3+4)^{2}}$$
2r=$$\sqrt{54}$$
r=$$\frac{3\sqrt{6}}{2}$$