Question

find the equation of the plane through (4,4,0) and perpendicular to the planes X+2y+2z=5 and 3x+3y+2z-8=0​

Answer 1

$\textbf{Given:}$

$\textsf{The plane passes through (4,4,0) and perpendicular to}$

$\textsf{x+2y+2z-5=0 and 3x+3y+2z-8=0}$

$\textbf{To find:}$

$\textsf{The equation of the plane}$

$\textbf{Solution:}$

$\textbf{Formula used:}$

$\boxed{\begin{minipage}{8cm} \\\\\textsf{Equation of the plane passes through a point and}\\\\\textsf{perpendicular two planes can be written as}\\\\\mathsf{\;\;\;\;\;\;\;\left|\begin{array}{ccc}x-x_1&y-y_1&z-z_1\\l_1&m_1&n_1\\l_2&m_2&n_2\end{array}\right|=0}\\\\ \end{minipage}}$

$\mathsf{Here,}$

$\mathsf{(x_1,y_1,z_1)=(4,4,0)}$

$\mathsf{(l_1,m_1,n_1)=(1,2,2)}$

$\mathsf{(l_2,m_2,n_2)=(3,3,2)}$

$\textsf{Equation of the required plane is}$

$\mathsf{\left|\begin{array}{ccc}x-x_1&y-y_1&z-z_1\\l_1&m_1&n_1\\l_2&m_2&n_2\end{array}\right|=0}$

$\mathsf{\left|\begin{array}{ccc}x-4&y-4&z-0\\1&2&2\\3&3&2\end{array}\right|=0}$

$\mathsf{\left|\begin{array}{ccc}x-4&y-4&z\\1&2&2\\3&3&2\end{array}\right|=0}$

$\textsf{Expanding along first row, we get}$

$\mathsf{(x-4)(4-6)-(y-4)(2-6)+z(3-6)=0}$

$\mathsf{(x-4)(-2)-(y-4)(-4)+z(-3)=0}$

$\mathsf{(x-4)(-2)-(y-4)(-4)+z(-3)=0}$

$\mathsf{-2(x-4)+4(y-4)-3z=0}$

$\mathsf{-2x+8+4y-16-3z=0}$

$\mathsf{-2x+4y-3z-8=0}$

$\implies\boxed{\mathsf{2x-4y+3z+8=0}}$

$\textbf{Find more:}$

Find vector equation of plane which passes through the points (3,2,1) and (0,1,7) and is parallel to line r=2i-j+k+l(i-j-k)

https://brainly.in/question/8252354