Question

The equation of a plane which passes through the point of intersection of lines $\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{2}$, and $\frac{x-3}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ and at greatest distance from point $(0,0,0)$ is-

• A. $4x+3y+5z=25$
• B. $4x+3y+5z=50$
• C. $3x+4y+5z=49$
• D. $x+7y-5z=2$

Answer 1

The correct option is B $4x+3y+5z=50$

Any point on the first line is $P(3\lambda +1,\lambda +2,2\lambda +3)$
and on the second line is $Q(u+3,2u+1,3u+2)$
$P$ and $Q$ represent the same point if $\lambda =u=1$
And the point of intersection of the given line is $P(4,3,5)$
The plane given in (a),(b),(c) and (d) all pass through $P$.
The plane at greatest distance is one which is at a distance equalt to $OP$ from the origin.
So, the distance of the plane from origin is $\sqrt{4^2+3^2+5^2}=\sqrt{50}$
The equation of plane is $4x+3y+5z=50$