Question

Statement-$1$: The perpendicular distance from $(1,4,-2)$ to the line joining $(2,1,-2)$ , $(0,-5,1)$ is $\frac{5\sqrt{26}}{7}$

Statement-$2$: The perpendicular distance from a point $P$ to the line joining the points $A$, $B$ is $\frac{|\overline{AP}\times \overline{AB}|}{|\overline{AB}|}$

• A. Statement-$1$ is true, statement-$2$ is true, statement-$2$ is a correct explanation for statement-$1$.
• B. Statement-$1$ is true, statement-$2$ is true, statement-$2$ is not a correct explanation for statement-$1$.
• C. Statement-$1$ is true, statement-$2$ is false
• D. Statement-$1$ is false, statement-$2$ is true

Answer 1

The correct option is D Statement-$1$ is false, statement-$2$ is true

1. Line $=(2,1,-2)+t(2,6,-3)$

Vector to line from point $=(1+2t,-3+6t,-3t)$
$(1+2t,-3+6t,-3t)\cdot (2,6,-3)=0$
$2+4t-18+36t+9t=0$
$t=\frac{16}{49}$

Length$=\sqrt{(1+2t{)}^2+(6t-3{)}^2+(-3t{)}^2}$

$=\sqrt{49t^2-32t+10}$

$=\sqrt{\frac{(16{)}^2}{49}-\frac{-32\times 16}{49}+10}$

$=\frac{1}{7}\left(\sqrt{490-{16}^2}\right)$

$=\frac{3\sqrt{26}}{7}$

(Alternate Method) :-

We know that, the perpendicular distance of point $P$ from the line joining $A$ and $B$ is $\frac{|\overline{AP}\times \overline{AB}|}{|\overline{AB}|}$

Here, $P=(1,4,-2)$, $A=(2,1,-2)$ and $B=(0,-5,1)$

$\overline{AP}=P-A=(-1,3,0)$ and $\overline{AB}=B-A=(-2,-6,3)$

Then, $\frac{|\overline{AP}\times \overline{AB}|}{|\overline{AB}|}=\frac{|(-1,3,0)\times (-2,-6,3)|}{|(-2,-6,3)|}$

$=\frac{|(9,3,12)|}{\sqrt{4+36+9}}=\frac{\sqrt{81+9+144}}{\sqrt{49}}=\frac{\sqrt{234}}{7}=\frac{3\sqrt{26}}{7}$

Hence, Statement-$1$ is false and Statement 2 is true.