Question

Which of the following conditions, such that the line $\frac{x-p}{l}=\frac{y-q}{m}=\frac{z-r}{n}$ lies on the plane $Ax+By+Cz+D=0$ is/ are correct?
(i) $lp+mq+nr+D=0$ (ii) $Ap+Bq+Cr+D=0$ (iii) $Al+Bm+Cn=0$
Select the correct answer using the codes given.

• A. (i) only
• B. (i) and (ii)
• C. (i) and (iii)
• D. (ii) and (iii)

Answer 1

Answer: (D)

(ii) and (iii)

Given

$\Rightarrow \frac{x-p}{l}=\frac{y-q}{m}=\frac{z-r}{n}=t$ (let)

$\Rightarrow x=lt+p,y=mt+q,z=nt+r$

Putting these values of $x,y,z$ on the given plane equation, we get

$\Rightarrow A(lt+p)+B(mt+q)+C(nt+r)$

$\Rightarrow Ap+Bq+Cr+D+t(Al+Bm+Cn)=0$

For this equation to satisfy

$\Rightarrow Ap+Bq+Cr+D=0,Al+Bm+Cn=0$

Hence, answer is option $D$