Question

Write the equation of the sphere in standard form. x2 + y2 + z2 + 8x − 6y + 4z + 20 = 0

Answer 1

Answer:

$(x+4)^2 + (y-3)^2 + (z+2)^2 - 3^2 = 0$

Step-by-step explanation:

You have to find a, b, c, r so that the polynomial expands to match the given equation.

$(x-a)^2 + (y-b)^2 + (z-c)^2 - r^2 =\\x^2-2ax+a^2+\\y^2-2by+b^2+\\z^2-2cz+c^2-r^2 =\\ x^2 + y^2 + z^2 + 8x - 6y + 4z + 20$

$(a^2+b^2+c^2-r^2-20)+\\x(-8-2a)+y(6-2b)+z(-4-2c)=0$

$-8-2a=0, 6-2b=0, -4-2c=0\\a=-4, b=3, c=-2$

$(-4)^2+3^2+(-2)^2-r^2-20 = 0$

$16+9+4-20=r^2$

$r^2=9$

$r=3$

$(x+4)^2 + (y-3)^2 + (z+2)^2 - 3^2 = 0$