Question

The equation of the palne passing through the line of intersection of the planes $x+2y+3z-5=0$ and $3x+2y-z+1=0$ and cutting off equal intercepts on the $OX$ and $OZ$ axes is

• A. $2x+5y-5z-9=0$
• B. $5x-2y-5z+9=0$
• C. $5x+2y+5z-9=0$
• D. $5x+5y-2z+9=0$

Answer 1

Answer: (C)

$5x+2y+5z-9=0$

Any plane passing through the line of intersection of the given planes is

$(x+2y+3z-5)+\lambda (3x-2y-z+1)=0$

$\Rightarrow (1+3\lambda )x+(2-2\lambda )y+(3-\lambda )z+(-5+\lambda )=0$ ...$\left(1\right)$

Intercept on $x$-axis is $\frac{5-\lambda }{1+3\lambda }$ (Putting $x=y=z=0$)

Intercept on $y$-axis is $\frac{5-\lambda }{3-\lambda }$

Given: $\frac{5-\lambda }{1+3\lambda }=\frac{5-\lambda }{3-\lambda }\Rightarrow \lambda =\frac{1}{2}$

$[\lambda =5$ is not admissible as in taht case each intercept is zero$]$

Put $\lambda =\frac{1}{2}$ in $\left(1\right)$, we get the equation of the required plane as $5x+2y+5z-9=0$