Question

Find the equation of the plane passing through the line if intersection of the planes x+y-28=0 and x-y+2z=0 and which arr at a distance of root 26 units from point (2,0,-1)

Answer 1

$$\begin{gathered} P1:x+y-28=0 \\ P2:x-y+2z=0 \\ A=\left(2,0,-1\right) \\ EquationofplainPpas\sin gthroughintersectionofP1andP2canbegivenbyfamilyofplains \\ P:x+y-28+\lambda \left(x-y+2z\right)=0 \\ P:\left(\lambda +1\right)x-\left(\lambda -1\right)y+2\lambda z-28=0 \\ Dis\tan ceofapoint(a,b,c)\hspace{0.17em}fromaplainAx+By+Cz+d=0isgivenby \\ d=\left|\frac{Aa+Bb+Cc+d}{\sqrt{A^2+B^2+C^2}}\right| \\ \sqrt{26}=\left|\frac{2\left(\lambda +1\right)-0-2\lambda -28}{\sqrt{{\left(\lambda +1\right)}^2+{\left(\lambda -1\right)}^2+4{\lambda }^2}}\right| \\ \sqrt{26}=\left|\frac{-26}{\sqrt{{\left(\lambda +1\right)}^2+{\left(\lambda -1\right)}^2+4{\lambda }^2}}\right| \\ 1=\left|\frac{\sqrt{26}}{\sqrt{{\left(\lambda +1\right)}^2+{\left(\lambda -1\right)}^2+4{\lambda }^2}}\right| \\ 2{\lambda }^2+2+4{\lambda }^2=26 \\ 6{\lambda }^2=24 \\ {\lambda }^2=4 \\ \lambda =\pm 2 \\ Equationofplaincanbe \\ -x+3y-4z-28=0 \\ or \\ 3x-y+4z-28=0 \end{gathered}$$