The distance of the point ( 3, 4, 5) from X- axis is

let L be the foot of perpendicular from the point ( 3, 4, 5) to X axis ,then coordinate of L will be (3,0,0) [ because on X axis y and z coordinate are zero]

then distance of the point ( 3, 4, 5) from X- axis i.e. from L (3,0,0) is given by =

As we know that if a lines makes angles * a *,

** In our case line is X-axis itself which we know makes angle of 0 , 90 , 90 with ** X-axis , Y-axis and Z-axis respectively then direction cosine will be

<cos 0 , cos 90, cos 90>

= < 1 0 0 >

The centroid is the point of concurrency of the medians of the triangle.it is a point of centre of gravity of triangle

**Concept used - Parellel lines have same direction cosines**

* Explanation *We know Z axis is perpendicular to XY plane and direction cosines of Z axis are <0,0,1>

and as any normal to XY plane will be parallel to Z axis and parellel lines have same direction cosines so answer is <0,0,1>

hence the lines will be 8

the numbers 3, 4 , 5 can be direction ratio of any line these not satisfying any other option

We know that if * l , m , n* are direction cosines of a lines then

so, k2 + k2 +k2 = 1

3 k2 = 1

k2 = 1/3

taking squareroot on both sides , we have

Lines which lie in same plane can be parallel, coincident or intersecting. Skew lines are those lines which lie in different planes and never meet to each other

we know that general equation of sphere is

since sphere passing through the points ( 4 ,3 , 0 ) , ( 0 , 4 , 3 ) ,( 0 , 5 , 0 ) and ( 4 , 0 , 3 ) so putting these values one by one in given equation , we get

16+ 9+ 8g + 6f + c=0 =>8g +6f +c = -25 -------------1)

16 +9 +8f +6h +c=0 =>8f + 6h +c= -25 -----------2)

25 + 10f +c = 0 -----------------------3)

16+ 9+ 8g + 6h +c=0 =>8g +6h +c= -25 -------------4)

by 2) - 4 ) we have f = g

using this in 1) we have 16f +c = -25 -------5)

Now solving 3) and 5) we get f=0

using value of f we have g= 0 , h= 0 , c= -25

Now radius = = =5

The vertices of a quadrilateral are A( 4,7 ,8) , B( 2, 3,4 ) ,C (- 1 , -2 , 1 ) and D(1 , 2, 5 )

Here opposite sides of quadrilateral ABCD are equal , so it may be parallelogram or rectangle

Here length of diagonals AC and BD are different so ABCD is a parrallelogram