Question

Assertion :The equation of the plane through the intersection of the planes $x+y+z=6$ and $2x+3y+4z+5=0$ and the point $(4,4,4)$ is $29x+23y+0z=276$ Reason: Equation of the plane through the line of intersection of the planes $P_1=0$ and $P_2=0$ is $P_1+\lambda P_2=0(\lambda \ne 0)$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is incorrect but Reason is correct
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is incorrect but Reason is correct

$P_1:x+y+z-6=0$

$P_2:2x+3y+4z+5=0$

Equation of plane passing through the intersection of plane $P_1$ and $P_2$ is

$P_1+\lambda P_2=0$ $(\lambda \ne 0)$

$x+y+z-6+\lambda (2x+3y+4z+5)=0$

$(1+2\lambda )x+(1+3\lambda )y+(1+4\lambda )z-6+5\lambda =0$

since these planes are passes through $(4,4,4)$, therefore,

$(1+2\lambda )4+(1+3\lambda )4+(1+4\lambda )4-6+5\lambda =0$

$\lambda =-\frac{6}{41}$

thus required equation of the plane is

$\left(1-\frac{12}{41}\right)x+\left(1-\frac{18}{41}\right)y+\left(1-\frac{24}{41}\right)z-6-\frac{30}{41}=0$

$29x+23y+17z=276$

Therefore, the assertion is wrong and the reason is correct.