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Chapter 14: Q. 5 (page 1119)

Compute a general formula for $$dS$$ for any plane $$ax +by+cz = k$$ if $$c \neq 0$$.

Short Answer

Expert verified

The computed formula for dS for any plane $$ax +by+cz = k$$ if $$c \neq 0$$ is $$\sqrt{\frac{a^{2}+b^{2}+c^{2}}{c^{2}}}$$

Step by step solution

01

Step 1. Given Information

Any plane $$ax +by+cz = k$$, where $$c \neq 0$$.

02

Step 2. Explanation

We get, $$dS=\sqrt{(\frac{\partial z}{\partial x})^{2}+\frac{\partial z}{\partial y})^{2}+1}dA$$ -----(1)

Also, the given plane is, $$ax +by+cz = k$$

$$\implies z=\frac{k}{c}+\frac{ax}{c}-\frac{by}{c}$$

Here, $$dS$$ will become,

$$dS=\sqrt{\frac{a^{2}}{c^{2}}+\frac{b^{2}}{c^{2}}+1}$$

$$\implies dS=\sqrt{\frac{a^{2}+b^{2}+c^{2}}{c^{2}}}$$

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