91Ó°ÊÓ

In three-dimensional geometry, a sphere is a set of points that are all the same distance from a fixed point, known as the center. This distance is called the radius. The general equation of a sphere in three-dimensional space is given by \[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\] where - \((h, k, l)\) represent the coordinates of the center of the sphere, - \(r\) is the radius of the sphere.In the problem's original form, we had the equation \(x^2 + y^2 + z^2 - 14x + 6y - 8z + 10 = 0\).To convert this to the standard equation of a sphere, we completed the square for each variable, revealing the sphere's center and radius.

Completing the square is a mathematical technique used to transform a quadratic expression into a perfect square trinomial. This method is essential for converting the general equation of a circle or sphere into its standard form. In the given problem, the equation had terms like \(x^2 - 14x\), \(y^2 + 6y\),and \(z^2 - 8z\).To complete the square:- For \(x\), find half of \(-14\) (which is \(-7\)) and square it to get \(49\). This makes \(x^2 - 14x\) into \((x-7)^2 - 49\).- For \(y\), half of \(6\) is \(3\), and squaring gives \(9\), so \(y^2 + 6y\) becomes \((y+3)^2 - 9\).- For \(z\), half of \(-8\) is \(-4\), and squaring yields \(16\). Hence, \(z^2 - 8z\) turns into \((z-4)^2 - 16\).These adjustments allow us to simplify and rearrange the equation into a more recognizable form.

Graphing in three-dimensional space can be a challenging but rewarding task, especially when it involves shapes like spheres. To graph a sphere from its equation, it is essential to first identify the center and radius from the standard form equation.For the sphere determined by our exercise:- Identify the center at \((7, -3, 4)\),- Recognize the radius \(8\), since \(64 = 8^2\).Once these elements are known, you can proceed with sketching:

• Begin by drawing a three-dimensional coordinate system.
• Plot the center of the sphere at the identified coordinates.
• Use the radius to determine the size of the sphere, ensuring symmetry in all directions from the center.

By carefully analyzing and plotting each element, creating a clear and accurate representation of a sphere in three-dimensional space becomes attainable.