Question

find the partial differential equation of all planes which are at a constant distance a from origin​

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

The required differential equation is

$z = px + qy \pm a \sqrt{p^2 + q^2 + 1}$

Step-by-step explanation:

Let us assume that the required equation of a plane is

z = lx + my + nlx + my − z + n = 0    -(i)

Now the plane (i) is at a constant distance 'd' from the origin

$\therefore d = \frac{|n|}{\sqrt{l^2 + m^2 + 1} } \\\\\implies d = \frac{\pm n}{\sqrt{l^2 + m^2 + 1} }\\\\Here,\ p = \frac{|ax_1 +by_1 +cz_1 + d|}{\sqrt{a^2 + b^2 + c^2} }\\\\\implies n = \frac{\pm n}{\sqrt{l^2 + m^2 + 1} }$

Now, the equation (i) becomes,

$lx + my - z \pm a \sqrt{l^2 + m^2 + 1} =0$      -(ii)

Now, differentiating the equation (ii) wrt to 'x', we get

$l - \frac{dz}{dx} = 0$ ⇒ p = l

Again, differentiating the equation (ii) wrt to 'y', we get

$m - \frac{dz}{zy} = 0$  ⇒ q = m

Now, the reduced equation (ii) is,

$px + qy - z \pm a \sqrt{p^2 + q^2 + 1} =0$

The required differential equation is

$z = px + qy \pm a \sqrt{p^2 + q^2 + 1}$

To learn more about the differential equation, click on the link below:

https://brainly.in/question/54118694

To learn more about differentiation, click on the link below:

https://brainly.in/question/758525

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