3D Geometry - 6
Introduction
Q.2. What are direction cosines of a line which is equally inclined to axes ? Ans: If







l2 + m2 + n2


















Q.3. Find direction cosines of line which is

Ans: Let l, m, n be d.c's line 1 to given line, d.c's are proportional to d.r's




Since line is

l




& 0 + 2mµ + nµ = 0

By cross multiplication,

l = 2R, m = - R & n = 2R
l2 + m2 + n2 = 1
4R2 + R2 + 4R2 = 1



Q.4. Prove that line x = ay + b, z = cy + d & x = a'y + b', z = c'y + d' are

Ans: Ist line is x = ay + b, z = cy + d






and IInd line

These lines are

aa' + |x| + cc' = 0

Dumb Question: For what value of k, lines


Ans:


and

Since these lines have point of intersection in common. then
(2



or 2



on solving (i) & (iii), we get

Substituting in (ii)
-



Q.6. Show that points




Ans:


length of



So, length of

Q.7. Find the point in which the plane;












Ans: eq. of line through point 2








Since line cuts plane
For one point,
(2












Equating coeff. of



m + n = 1, m - n = 4, - m + n =









Q.8. Show that line of intersection of planes






Ans: Note: Line of intersection of two planes will be


Now,

&

So, line is equally inclined to


Q.9. Projection of line segment on 3 axes 4, 5 & 13 respectively. Find length & direction cosines of line segment.
Ans:







=


Projection of

=


Projection of

=










Q.10. Find locus of mid point of chords of sphere r2 - 2



Ans: r2 - 2
















Dumb Question: Why






Ans: Since







Medium Type
Q.1. If a variable plane forms a tetrahedron of constant volume 27k3 with coordinate planes, find locus of centroid of tetrahedron.
Ans: sLet variable plane cuts coordinate axes at A(a, 0, 0), B(0, b, 0), C(0, 0, c)

Then, eq. of plane will be

Let P



then,






[Dumb Question: How this centroid tetrahedron OABC comes






Ans: Centroid of tetrahedrold

where (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) & (x4, y4, z4) are coordinate of tetrahedron.]
Volume of tetrahedron =


27k3 =






Required locus of P(




Q.2. Find vector eq. of straight line passing through intersection of plane




Ans: At points of intersection of two planes.

Since



-



