Question

Obtain partial differential equation of all spheres whose centre lies on z-axis with a given radius r

Answer 1

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Answer 2

Given :

Sphere whose center lies on z-axis.

To find :

Partial differential equation of sphere.

Solution :

The equation of sphere whose center lie on z-axis with radius r is :

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

(equation 1)

to find the partial differential equation of sphere,

firstly we are differentiating with respect to x.

$\frac{ \delta({x}^{2} + {y}^{2} + {(z - c)}^{2} )}{ \delta \: x} = \frac{ \delta {r}^{2} }{ \delta \: x} \: = 0$

$2x \: + \: 2(z - c)p = 0$

(equation 2)

where

$\frac{ \delta \: z}{ \delta \: x} = p$

then we are differentiating with respect to y.

$2y \: + \: 2(z - c)q = 0$

(equation 3)

where

$\frac{ \delta \: z}{ \delta \: y} = q$

by solving equation 2 and 3 :

$(z - c)p = - x \\ and \\ (z - c)q = - y$

dividing these we get :

$\frac{p}{q} = \frac{x}{y}$

it means

py = qx

So the desired partial differential equation is :

py - qx = 0