Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

$P_1:\ell x+my=0,P_2=z=0$ Plane through common line of $P_1$ and $P_2$ ${\mathrm{P}}_3:\ell \mathrm{x}+\mathrm{my}+\mathrm{nz}=0$ angle between $P_1$ and $P_3=\alpha$ $\therefore \cos \alpha ={\hat{n}}_1,{\hat{n}}_2$ $=\frac{{\ell }^2+{\mathrm{m}}^2}{\sqrt{{\ell }^2+{\mathrm{m}}^2}\sqrt{{\ell }^2+{\mathrm{m}}^2+{\mathrm{n}}^2}}=\sqrt{\frac{{\ell }^2+{\mathrm{m}}^2}{{\ell }^2+{\mathrm{m}}^2+{\mathrm{n}}^2}}$ $\Rightarrow {\cos }^2\alpha =\frac{{\ell }^2+{\mathrm{m}}^2}{{\ell }^2+{\mathrm{m}}^2+{\mathrm{n}}^2}$ $\Rightarrow {\mathrm{n}}^2=\left({\ell }^2+{\mathrm{m}}^2\right){\tan }^2\alpha$ $\Rightarrow \mathrm{n}=\pm \sqrt{{\ell }^2+{\mathrm{m}}^2}\tan \alpha$ ${\mathrm{P}}_3:\ell \mathrm{x}+\mathrm{my}\pm \sqrt{{\ell }^2+{\mathrm{m}}^2}$ tan $\alpha =0$