Question

The value of $a$, for which the point $A,B,C$ with position vectors $2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}$ and $a\hat{i}-3\hat{j}+\hat{k}$ respectively are the vertices of a right-angled triangle with $C=\frac{\pi }{2}$ are

• A. $-2$ and $-1$
• B. $-2$ and $1$
• C. $2$ and $-1$
• D. $2$ and $1$

Answer 1

The correct option is D $2$ and $1$

$A=2i-j+k=(2,-1,1)$

$B=i-3j-5k=(1,-3,-5)$

$C=ai-3j+k=(a,-3,1)$

(Image)

$c=\sqrt{1^2+2^2+6^2}$

$=\sqrt{36+4+1}$

$=c\sqrt{41}$

$b=\sqrt{{(a-2)}^2+2^2}$

$a=\sqrt{{(a-1)}^2+6^2}$

$\angle c=90°a^2+b^2=c^2$

${(a-1)}^2+6^2+{(a-2)}^2+2^2=41$

${(a-1)}^2+{(a-2)}^2=1$

$a^2+1-2a+a^2+4-4a=1$

$2a^2-6a+4=0$

$a^2-3a+2=0$

$a^2-2a-a+2=0$

$a(a-2)-1(a-2)=0$

$a=1,2$