Question

A plane $\pi$ contains the line $L_1:\frac{y}{b}+\frac{z}{c}=1,x=0$ and is parallel to the line $L_2:\frac{x}{a}-\frac{z}{c}=1,y=0,$ then

• A. Equation of plane $\pi$ is $-\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
• B. Equation of plane $\pi$ is $\frac{x}{a}+\frac{y}{b}-\frac{z}{c}=1$
• C. if shortest distance between $L_1$ and $L_1$ is $\frac{1}{4},$ then $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=64$
• D. if shortest distance between $L_1$ and $L_1$ is $\frac{1}{4},$ then $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=192$

Answer 1

Answer: (A), (C)

The correct options are
A

Equation of plane $\pi$ is $-\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$

C

if shortest distance between $L_1$ and $L_1$ is $\frac{1}{4},$ then $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=64$

Equation of $L_1\equiv \frac{x}{0}=\frac{y}{-b}=\frac{z-c}{c}$

Equation of $L_2\equiv \frac{x}{a}=\frac{y}{0}=\frac{z+c}{c}$

$\therefore$ Direction of plane $\pi =\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 0& -b& c\\ a& 0& c\end{array}\right|$
$=-bc\hat{i}+ac\hat{j}+ab\hat{k}$

Equation of plane $\pi$ is

$-bc(x-0)+ac(y-0)+ab(z-c)=0$

$-\frac{x}{a}+\frac{y}{b}+\frac{z}{c}-1=0$

$\therefore$ Distance between $L_1$ and $L_2$ is

$|\frac{(0,0,2c).(bc,ac,-ab)}{\sqrt{b^2{c}^2+a^2{c}^2+a^2{b}^2}}|\therefore \frac{1}{4}=|\frac{2abc}{\sqrt{b^2{c}^2+a^2{c}^2+a^2{b}^1}}|\Rightarrow \frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=64$