Write the equation of the sphere in standard form calculator

Asked by admin @ 28/10/2021 in Mathematics viewed by 333 People

Write the equation of the sphere in standard form. 2x2 + 2y2 + 2z2 = 12x − 24z + 1

Answered by admin @ 28/10/2021

The standard equation of a sphere with center C(a, b, c) and radius r is:

$(x-a)^{2}+ (y-b)^{2}+ (z-c)^{2}=r^{2}$

We have the equation:

$2x^2+2y^2+2z^2=12x-24z+1$

which can be regrouped to fit the standard form, using completing the square as follows:

first divide both sides by 2

$x^2+y^2+z^2=6x-12z+ \frac{1}{2} \\\\(x^2-6x)+y^2+(z^2+12z)= \frac{1}{2}\\\\(x^2-2\cdot3\cdot x)+y^2+(z^2+2 \cdot6 \cdot z)= \frac{1}{2}\\\\(x^2-2\cdot3\cdot x+3^2)-3^2+y^2+(z^2+2 \cdot6 \cdot z+6^2)-6^2= \frac{1}{2}\\\\(x-3)^2+y^2+(x+6)^2=9+36+0.5\\\\(x-3)^2+y^2+(x+6)^2=45.5\\\\\(x-3)^2+y^2+(x+6)^2=( \sqrt{45.5} )^2$

Answer: $(x-3)^2+y^2+(x+6)^2=( \sqrt{45.5} )^2$

Write the equation of the sphere in standard form calculator

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