# Mathematics-I (3110014)

BE | Semester-1   Winter-2019 | 17-01-2020

## Q1) (a)

#### Find the equations of the tenagent plane and normal line to the surface ${x}^{2}+{y}^{2}+{z}^{2}=3$ at the point $\left(1,1,1\right)$.

$\mathrm{Here,}f\left(x,y,z\right)={x}^{2}+{y}^{2}+{z}^{2}-3$ $\mathrm{and}$ .

$\mathrm{Now,}$

• $\frac{\partial \mathrm{f}}{\partial \mathrm{x}}=\frac{\partial }{\partial \mathrm{x}}\left({\mathrm{x}}^{2}+{\mathrm{y}}^{2}+{\mathrm{z}}^{2}-3\right)=2\mathrm{x}$

$⟹{\left(\frac{\partial \mathrm{f}}{\partial \mathrm{x}}\right)}_{\mathrm{p}}=2\left(1\right)=2$

• $\frac{\partial \mathrm{f}}{\partial \mathrm{y}}=\frac{\partial }{\partial \mathrm{y}}\left[{\mathrm{x}}^{2}+{\mathrm{y}}^{2}+{\mathrm{z}}^{2}-3\right]=2\mathrm{y}$

$⟹{\left(\frac{\partial \mathrm{f}}{\partial \mathrm{y}}\right)}_{\mathrm{p}}=2\left(1\right)=2$

• $\frac{\partial \mathrm{f}}{\partial \mathrm{z}}=\frac{\partial }{\partial \mathrm{z}}\left[{\mathrm{x}}^{2}+{\mathrm{y}}^{2}+{\mathrm{z}}^{2}-3\right]=2\mathrm{z}$

$⟹{\left(\frac{\partial \mathrm{f}}{\partial \mathrm{z}}\right)}_{\mathrm{p}}=2\left(1\right)=2$

$⟹2\mathrm{x}-2+2\mathrm{y}-2+2\mathrm{z}-2=0$

$⟹2\mathrm{x}+2\mathrm{y}+2\mathrm{z}-6=0$

$⟹2\mathrm{x}+2\mathrm{y}+2\mathrm{z}=6$

$⟹\mathbf{x}\mathbf{=}\mathbf{y}\mathbf{=}\mathbf{z}$