Amit M Agarwal Solutions for Chapter: Three Dimensional Coordinate System, Exercise 10: Three Dimensional Coordinate System Exercise 7

Find the point on the line $\frac{x+2}{3}=\frac{y+1}{2}=\frac{z-3}{2}$ at a distance of $3\sqrt{2}$ from the point $\left(1,2,3\right)$

A line passes through $\left(2,-1,3\right)$ and is perpendicular to the lines $\overrightarrow{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda \left(2\hat{i}-2\hat{j}+\hat{k}\right)$ and $\overrightarrow{r}=\left(2\hat{i}-\hat{j}-3\hat{k}\right)+\mu \left(\hat{i}+2\hat{j}+2\hat{k}\right)$ obtain its equation.

Find the equations of the two lines through the origin which intersect the line $\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at angle of $\frac{\pi }{3}$ each

Vertices $B$ and $C$ of $\Delta ABC$ lie along the line $\frac{x+2}{2}=\frac{y-1}{1}=\frac{z-0}{4}.$ Find the area of the triangle given that $A$ has coordinates $\left(1,-1,2\right)$ and line segment $BC$ has length $5$.

A point $P$ moves on the plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ which is fixed. The plane through $P$ perpendicular to $OP$ meets the axes in $A,B \& C$. The planes through $A,B,C$ parallel to $yz, zx, xy$ planes intersect in $Q$. Prove that if the axis is rectangular, then the locus of $Q$ is $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{ax}+\frac{1}{by}+\frac{1}{cz}$.

Prove that the distance of the point of intersection of the line $\frac{x–2}{3}=\frac{y+1}{4}=\frac{z–2}{12}$ and the plane $x-y+z=5$ from the point $\left(-1,-5,-10\right)$ is $13$.

Find the equation of the plane through the intersection of the planes $x+3y+6=0 \& 3x–y–4z=0$, whose perpendicular distance from the origin is unity.

Find the equation of the image of the plane $x–2y+2z–3=0$ in the plane $x+y+z–1=0$.