Question

The equation of plane through line of intersection of plane $x+y+z-4=0$ and $2x-3y+z+5=0$ and whose x intercept equal to unity

• A. $13x-2y=10z=13$
• B. $x+y-4z=1$
• C. $3x-9y=4z=3$
• D. $2x-y-z=2$

Answer 1

Answer: (A)

$13x-2y=10z=13$

$\begin{array}{c}Equationofaplanethroughthelineofinter\sec tionofplanex+y+z-4=0\\ and2x-3y+z+5=0\\ \left(x+y+z-4\right)+\lambda \left(2x-3y+z+5\right)=0\\ x\left(1+2\lambda \right)+y\left(1-3\lambda \right)+z\left(1+\lambda \right)-4+5\lambda =0\\ x\left(2\lambda +1\right)+y\left(1-3\lambda \right)+z\left(\lambda 4\right)=4-5\lambda \\ \frac{x}{\frac{4-5\lambda }{2\lambda +1}}+\frac{y}{\frac{4-5\lambda }{1-3\lambda }}+\frac{z}{\frac{4-5\lambda }{\lambda +1}}=1\\ \frac{4-5\lambda }{2\lambda +1}=1\\ 4-5\lambda =2\lambda +1\\ 3=7\lambda \\ \lambda =\frac{3}{7}\\ x\left(1+\frac{6}{7}\right)+y\left(1-\frac{9}{7}\right)+z\left(1+\frac{3}{7}\right)-4+\frac{15}{7}=0\\ 13x-2y+10z-13=0\\ Hence,thecorrectoptionis\left(A\right)\end{array}$