Embibe Experts Solutions for Chapter: Three Dimensional Geometry, Exercise 3: EXERCISE-3

The equation of the plane which has the property that the point $Q\left(5,4,5\right)$ is the reflection of point $P\left(1,2,3\right)$ through that plane, is $ax+by+cz=d$ where $a,b,c,d\in N$. Find the least value of $a+b+c+d$

If the angle between the planes given by $6x^2+4y^2-10z^2+3yz+4zx-11xy=0$ is ${\cos }^{-1}\left(k\right)$, then the value of '$k$' is equal to

Let the equation of the plane containing the line $x-y-z-4=0=x+y+2z-4$ and is parallel to the line of intersection of the planes $2x+3y+z=1$ and $x+3y+2z=2$ be $x+Ay+Bz+C=0$ Compute the value of $\left(\frac{A+B+C}{2}\right)$.

Through a point $P\left(f,g,h\right)$, a plane is drawn at right angles to $OP$ where '$O$' is the origin, to meet the coordinate axes in $A,B,C$. If the area of the triangle $ABC$ is $\frac{r^5}{\lambda fgh}$ where $OP=r$, then the value of $\lambda$ is

The position vectors of the four angular points of a tetrahedron $OABC$ are $\left(0,0,0\right);\left(0,0,2\right) ;\left(0,4,0\right)$ and $\left(6,0,0\right)$ respectively. A point $P$ inside the tetrahedron is at the same distance '$r$' from the four plane faces of the tetrahedron. The value of $\left(\frac{3r}{8}\right)$ is

If the line $\frac{x-1}{5}=\frac{y-2}{{\alpha }^2}=\frac{3\left(z-2\right)}{4}$ lie in the plane $3x-4y+5z=5$ then the value of $\left(\frac{65}{4{\alpha }^2}\right)$ is

$ABCD$ is a tetrahedron where $A\left(2,0,0\right), B\left(0,4,0\right)$ and $CD=\sqrt{14}$. The edge $CD$ lies on the line $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}$. If locus of centroid of tetrahedron is $\frac{x-\frac{3}{2}}{1}=\frac{y-y_1}{a}=\frac{z-z_1}{b}$, then the value of $\left(a+b+y_1+z_1\right)$ is

Given three point on $x-y$ plane as $O\left(0, 0\right), A\left(1, 0\right) \text{\&} B\left(-1,0\right)$. Point $P$ is moving on the plane satisfying the condition $\left(\overrightarrow{PA}·\overrightarrow{PB}\right)+3\left(\overrightarrow{OA}·\overrightarrow{OB}\right)=0$. If the maximum & minimum values of $\left|\overrightarrow{PA}\right|\left|\overrightarrow{PB}\right|$ are $M \text{\&} m$ respectively then the value of $M^2+m^2$ is