Lines r=(2 i-3 j+7 k)+ (2 i+p j+5 k) and r=(p i+2 j+3 k)+ (3 i-p … - Vidyadip

MCQ

Lines $\bar{r}=(2 \hat{i}-3 \hat{j}+7 \hat{k})+\lambda(2 \hat{i}+p \hat{j}+5 \hat{k})$ and $\bar{r}=(p \hat{i}+2 \hat{j}+3 \hat{k})+\mu(3 \hat{i}-p \hat{j}+p \hat{k})$ are perpendicular for all values of $\lambda$ and $\mu$ then $p$ is equal to

A
1, -6
B
1, 6
C
$-1,-6$
✓
$-1,6$

✓

Answer

Correct option: D.

$-1,6$

(D)
$a_1, b_1, c_1=2, p, 5$ and $a _2, b_2, c _2=3,- p , p$
Since, the given lines are perpendicular.
$\therefore \quad(2)(3)+p(-p)+(5)(p)=0$
$\Rightarrow 6- p ^2+5 p =0$
$\Rightarrow p ^2-5 p -6=0$
$\Rightarrow( p -6)( p +1)=0$
$\Rightarrow p=6$ or $p=-1$

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