Question

Let the points $A(a,b,c)$ and $B(a^{\prime },b^{\prime },c^{\prime })$ be at distances $r$ and $r^{\prime }$ from origin. The line $AB$ passes through origin when

• A. $\frac{a^{\prime }}{a}=\frac{b^{\prime }}{b}=\frac{c^{\prime }}{c}$
• B. $a{a}^{\prime }+b{b}^{\prime }+c{c}^{\prime }=r{r}^{\prime }$
• C. $a{a}^{\prime }+b{b}^{\prime }=c{c}^{\prime }=r^2+r{}^{\prime 2}$
• D. None of these

Answer 1

The correct option is A $\frac{a^{\prime }}{a}=\frac{b^{\prime }}{b}=\frac{c^{\prime }}{c}$

Equation of line $AB$ where $A$ is $(a,b,c)$ and $B$ is $(a,b,c)$

$\frac{x-a}{a-a}=\frac{y-b}{b-b}=\frac{-z-c}{c-c}=\lambda$

If it passes through origin $(0,0,0)$

then

$\frac{-a}{a-a}=\frac{-b}{b-b}=\frac{-c}{c-c}$

$\frac{a-a}{a}=\frac{b-b}{b}=\frac{c-c}{c}$

$\frac{1-a}{a}=\frac{1-b}{b}=\frac{1-c}{c}$

$\frac{a}{a}=\frac{b}{b}=\frac{c}{c}$

$A$ is correct