Answer:
(x + 5)² + (y - 4)² + z² = 109
Explanation:
the equation of a sphere in standard form is
(x - h)² + (y - k)² + (z - j)² = r²
(h, k, j ) are the coordinates of the centre and r is the radius
given
x² + y² + z² + 10x - 8y + 10 = 78 ( subtract 10 from both sides )
x² + y² + z² + 10x - 8y = 68 (collect the terms in x and y )
x² + 10x + y² - 8y + z² = 68
to complete the square on the x and y terms
add ( half the coefficient of the x/ y terms )² to both sides
x² + 2(5)x + 25 + y² + 2(- 4)y + 16 + z² = 68 + 25 + 16
(x + 5)² + (y - 4)² + z² = 109 ← in standard form