Embibe Experts Solutions for Chapter: Three Dimensional Geometry, Exercise 1: JEE Main - 10th April 2016

Let the line $L:\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}$ intersect the plane $2x+y+3z=16$ at the point $P$. Let the point $Q$ be the foot of perpendicular from the point $R\left(1,-1,-3\right)$ on the line $L$. If $\alpha$ is the area of triangle $PQR$. then ${\alpha }^2$ is equal to _____ .

Let $P$ be the plane, passing through the point $\left(1,-1,-5\right)$ and perpendicular to the line joining the points $\left(4,1,-3\right)$ and $\left(2,4,3\right)$. Then the distance of $P$ from the point $\left(3,-2,2\right)$ is

Let the plane $P:8x+{\alpha }_{1}y+{\alpha }_{2}z+12=0$ be parallel to the line $L:\frac{x+2}{2}=\frac{y-3}{3}=\frac{z+4}{5}$. If the intercept of $P$ on the $y$-axis is $1$ , then the distance between $P$ and $L$ is

The shortest distance between the lines $\frac{x-5}{1}=\frac{y-2}{2}=\frac{z-4}{-3}$ and $\frac{x+3}{1}=\frac{y+5}{4}=\frac{z-1}{-5}$ is

Let the image of the point $P\left(2,-1,3\right)$ in the plane $x+2y-z=0$ be $Q$. Then the distance of the plane $3x+2y+z+29=0$ from the point $Q$ is

Let the plane $P$ pass through the intersection of the planes $2x+3y-z=2$ and $x+2y+3z=6$, and be perpendicular to the plane $2x+y-z+1=0$. If $d$ is the distance of $P$ from the point $\left(-7,1,1\right)$, then $d^2$ is equal to :

Let $\alpha x+\beta y+\gamma z=1$ be the equation of a plane passing through the point $\left(3,-2,5\right)$ and perpendicular to the line joining the points $\left(1,2,3\right)$ and $\left(-2,3,5\right)$. Then the value of $\alpha \beta y$ is equal to _____ .

The point of intersection $C$ of the plane $8x+y+2z=0$ and the line joining the points $A\left(-3,-6,1\right)$ and $B\left(2,4,-3\right)$ divides the line segment $AB$ internally in the ratio $k:1$. If $a,b,c$ $\left(\right|a|,|b|,|c|$ are coprime) are the direction ratios of the perpendicular from the point $C$ on the line $\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}$, then $\left|a+b+c\right|$ is equal to _____ .