Q7Mathematics II

Question

Q.7. Find the centre and radius of the sphere $2 x^{2} + 2 y^{2} + 2 z^{2} - 3 x + 5 y - 4 z = 13$ .

Answer 1

Given the function containing arbitrary constants $a$ and $b$ :

z = a (x + y) + b

To systematically eliminate the arbitrary constants, we strictly differentiate $z$ partially with respect to $x$ (holding $y$ completely constant):

p = \frac{\partial z}{\partial x} = a (1 + 0) + 0 = a

Next, we strictly differentiate $z$ partially with respect to $y$ (holding $x$ completely constant):

q = \frac{\partial z}{\partial y} = a (0 + 1) + 0 = a

Since both partial derivatives $p$ and $q$ mathematically equal the exact same constant $a$ , we rigorously equate them completely:

p = q

Question