Browse Questions

# The equation of the plane through origin and the line of intersection of the planes $\overrightarrow r.\overrightarrow a=\lambda\:\:and\:\:\overrightarrow r.\overrightarrow b=\mu$ is ?

$\begin{array}{1 1} (a)\:\overrightarrow r.(\lambda\overrightarrow a-\mu\overrightarrow b)=0\:\qquad\:(b)\:\overrightarrow r.(\lambda\overrightarrow b-\mu\overrightarrow a)=0\:\qquad\:(c)\:\overrightarrow r.(\lambda\overrightarrow a+\mu\overrightarrow b)=0\:\qquad\:(d)\:\overrightarrow r.(\lambda\overrightarrow b+\mu\overrightarrow a)=0 \end{array}$

Toolbox:
• Equation of any plane passing through the line of intersection of the planes $P_1=0$ and $P_2=0$ is given by $P_1+\alpha P_2=0$
Eqn of the plane through the line of intersection of the planes $\overrightarrow r.\overrightarrow a-\lambda=0 \:\:and\:\:\overrightarrow r.\overrightarrow b-\mu=0$ is
$(\overrightarrow r.\overrightarrow a-\lambda)+\alpha (\overrightarrow r.\overrightarrow b-\mu)=0$...(i)
But given that this plane passes through origin.
$\therefore \:\overrightarrow r=0$, satisfies the ewquation of the plane.
$\Rightarrow\:\alpha=-\large\frac{\lambda}{\mu}$
Substituting the value of $\lambda$ in (i) we get the required equation of the plane as
$\overrightarrow r.(\mu \overrightarrow a-\lambda \overrightarrow b)=0$