Question

The equation of line passing through $(2,-3,5)$ and making congruent angles with all three co-ordinate axes is _______ ($k\in R$)

• A. $F=(2,-3,5)+k(1,1,1),k\in R$
• B. $F=(2,-3,5)+k(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}),k\in R$
• C. $F=(2,-3,5)+k(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}),k\in R$
• D. $F=(2,-3,5)+k(-1,1,1),k\in R$

Answer 1

Answer: (A)

$F=(2,-3,5)+k(1,1,1),k\in R$

$\therefore \alpha =\beta =\gamma$
Now ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
$\therefore 3{\cos }^2\alpha =1$
$\because \alpha =\beta =\gamma =1$
$\therefore {\cos }^2\alpha =\frac{1}{\sqrt{3}}$
$\therefore (l,m,n)=\overline{l}=(\cos \alpha ,\cos \beta ,\cos \gamma )=\pm \frac{1}{\sqrt{3}}(1,1,1)$
$\therefore$ Direction of a line $\overline{l}=(1,1,1)$
$\therefore$ Equation of a line is $\overline{r}=\overline{a}+k\overline{l};k\in R$
$\overline{r}=(2,-3,5)+k(1,1,1),k\in R$