Class 12

Math

3D Geometry

Three Dimensional Geometry

The distance between the parallel planes $2x−3y+6z=5$ and $6x−9y+18z+20=0$, is?

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Value ofλ such that the linex−12=y−13=z−1λIs perpendicular to normal to the planer⃗ .(2i⃗ +3j⃗ +4k⃗ )=0 is

Determine whether the following pair of lines intersect or not. (1) $r=i^−5j^ +λ(2i^+k^);r=2i^−j^ +μ(i^+j^ −k^)$ (2) $r=i^+j^ −k^+λ(3i^−j^ );r=4i^−k^+μ(2i^+3k^)$

Find the angle between the line whose direction cosines are given by $l+m+n=0and2l_{2}+2m_{2}−n_{2}−0.$

A mirror and source of light are situated at the origin O and a point on OX respectively. A ray of light from the source strikes the mirror and is reflected. If the DRs of the normal to the plane of mirror are $1,−1,1,$ then DCs for the reflacted ray are :

Find the equations of the bisectors of the angles between the planes $2x−y+2z+3=0and3x−2y+6z+8=0$ and specify the plane which bisects the acute angle and the plane which bisects the obtuse angle.

The extremities of a diameter of a sphere lie on the positive y- and positive z-axes at distance 2 and 4, respectively. Show that the sphere passes through the origin and find the radius of the sphere.

A sphere of constant radius $k,$ passes through the origin and meets the axes at $A,BandC˙$ Prove that the centroid of triangle $ABC$ lies on the sphere $9(x_{2}+y_{2}+z_{2})=4k_{2}˙$

Find the acute angle between the lines $lx−1 =my+1 =n1 and=mx+1 =ny−3 =lz−1 wherel>m>n,andl,m,n$ are the roots of the cubic equation $x_{3}+x_{2}−4x=4.$