Miscellaneous Exercise — Introduction to Three Dimensional Geometry

1. M.14 marksNCERT Class 11 Mathematics, Introduction to Three Dimensional Geometry, Reprint 2026-27

   Three vertices of a parallelogram ABCD are A(3,-1,2), B(1,2,-4), and C(-1,1,2). Find the coordinates of the fourth vertex D.

   Hint. The diagonals of a parallelogram bisect each other. In ABCD, the diagonals are AC and BD, so set their midpoints equal.

   Midpoint of AC = ((3-1)/2,(-1+1)/2,(2+2)/2)=(1,0,2). Let D=(x,y,z). Midpoint of BD = ((1+x)/2,(2+y)/2,(-4+z)/2). Setting this equal to (1,0,2): (1+x)/2=1 gives x=1; (2+y)/2=0 gives y=-2; (-4+z)/2=2 gives z=8.

   ✦ Working through each part gives: D = (1, -2, 8).

2. M.25 marksNCERT Class 11 Mathematics, Introduction to Three Dimensional Geometry, Reprint 2026-27

   Find the lengths of the medians of the triangle with vertices A(0,0,6), B(0,4,0), and C(6,0,0).

   Hint. A median joins a vertex to the midpoint of the opposite side. Find all three midpoints first, then apply the distance formula from each opposite vertex.

   Midpoint of BC = (3,2,0); median from A has length sqrt[(3-0)^2+(2-0)^2+(0-6)^2]=sqrt[9+4+36]=sqrt(49)=7. Midpoint of CA = (3,0,3); median from B has length sqrt[(3-0)^2+(0-4)^2+(3-0)^2]=sqrt[9+16+9]=sqrt(34). Midpoint of AB = (0,2,3); median from C has length sqrt[(0-6)^2+(2-0)^2+(3-0)^2]=sqrt[36+4+9]=sqrt(49)=7.

   ✦ Working through each part gives: The medians have lengths 7 (from A), sqrt(34) (from B), and 7 (from C).

3. M.34 marksNCERT Class 11 Mathematics, Introduction to Three Dimensional Geometry, Reprint 2026-27

   If the origin is the centroid of the triangle PQR with vertices P(2a,2,6), Q(-4,3b,-10), and R(8,14,2c), find the values of a, b, and c.

   Hint. Set the average of the three vertices' coordinates equal to the origin (0,0,0), one coordinate at a time.

   For the x-coordinates: (2a-4+8)/3=0, so 2a+4=0, giving a=-2. For the y-coordinates: (2+3b+14)/3=0, so 3b+16=0, giving b=-16/3. For the z-coordinates: (6-10+2c)/3=0, so 2c-4=0, giving c=2.

   ✦ Working through each part gives: a = -2, b = -16/3, c = 2.

4. M.44 marksNCERT Class 11 Mathematics, Introduction to Three Dimensional Geometry, Reprint 2026-27

   If A and B are the points (3,4,5) and (-1,3,-7) respectively, find the equation of the set of points P such that PA^2+PB^2=k^2, where k is a constant.

   Hint. Write out PA^2 and PB^2 in full, add them together, and simplify — this mirrors the textbook's own Example 6, which uses the same two points with 2k^2 on the right instead of k^2.

   PA^2=(x-3)^2+(y-4)^2+(z-5)^2=x^2+y^2+z^2-6x-8y-10z+50. PB^2=(x+1)^2+(y-3)^2+(z+7)^2=x^2+y^2+z^2+2x-6y+14z+59. Adding: PA^2+PB^2=2x^2+2y^2+2z^2-4x-14y+4z+109. Setting this equal to k^2: 2x^2+2y^2+2z^2-4x-14y+4z+109=k^2, which rearranges to 2x^2+2y^2+2z^2-4x-14y+4z=k^2-109.

   ✦ Working through each part gives: 2x^2 + 2y^2 + 2z^2 - 4x - 14y + 4z = k^2 - 109.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 11 Mathematics textbook, Reprint 2026-27 (kemh111.pdf) — Exercise 11.1 (4 questions), Exercise 11.2 (5 questions), plus the chapter's Miscellaneous Exercise (4 questions), 13 questions total, the shortest solutions file in this project to date, matching the book's own shortest chapter (9 pages, only two numbered exercises). Confirmed against the CBSE curriculum PDF that the section formula is formative-only for this chapter and confirmed via full-text search that it is completely absent from the book (zero hits for 'section formula', 'internal division', or 'ratio m'). The old stub had fabricated a third numbered exercise ('Exercise 11.3', 5 questions on the section formula) that does not exist in the book at all — removed. Midpoint and centroid, while never given a named boxed formula in the book, are confirmed as genuinely required techniques via the book's own Example 7 (parallelogram diagonals bisecting each other), Example 9 (centroid used to find a missing vertex), and the Miscellaneous Exercise's own Q1 (parallelogram's fourth vertex) and Q2 (median lengths).. Questions are referenced from the NCERT textbook for identification.