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The Vault
RowQ
The Vault
CBSE Class 11 Maths · 10 questions · 24 marks
Real space has three directions, not two, so this short chapter extends coordinate geometry from the plane to space by adding a third axis. You will locate points using an (x, y, z) triple, measure the distance between two such points, and find the point that divides a segment in a given ratio — the same ideas you know from 2D coordinate geometry, just carried one dimension further. It is the launchpad for vectors and 3D geometry in Class 12.
The point (0, -5, 3) lies on:
Answer
It lies on the YZ-plane. The YZ-plane consists of all points with x-coordinate 0. Here x = 0, y = -5, z = 3, so the point satisfies x = 0 and lies on the YZ-plane.
The distance between the points (1, 2, 3) and (4, 6, 3) is:
Answer
The distance is 5. Distance = √[(4-1)²+(6-2)²+(3-3)²] = √[9+16+0] = √25 = 5.
The distance of the point (3, -4, 12) from the origin is:
Answer
The distance is 13. Distance from origin = √(x²+y²+z²) = √(9+16+144) = √169 = 13.
The coordinates of the midpoint of the segment joining (2, -3, 6) and (8, 5, -2) are:
Answer
The midpoint is (5, 1, 2). Midpoint = [(2+8)/2, (-3+5)/2, (6-2)/2] = (10/2, 2/2, 4/2) = (5, 1, 2).
Assertion (A): The point (0, 0, 7) lies on the z-axis. Reason (R): Every point on the z-axis has both its x- and y-coordinates equal to zero.
Answer
Both A and R are true and R is the correct explanation of A. The point (0,0,7) has x=0 and y=0, matching the description of points on the z-axis, so Assertion A is true. Reason R states this exact defining property of the z-axis, and checking it is precisely how we confirm A, so R correctly explains A.
Find the coordinates of the point which divides the segment joining (4, -3, 2) and (1, 6, 8) internally in the ratio 2:1.
Answer
Using the section formula with m=2, n=1, P(4,-3,2), Q(1,6,8): x = (2×1+1×4)/(2+1) = (2+4)/3 = 2. y = (2×6+1×(-3))/3 = (12-3)/3 = 3. z = (2×8+1×2)/3 = (16+2)/3 = 6. So the point is (2, 3, 6).
Show that the points A(1, 2, 3), B(3, 8, 11) and C(4, 5, -3) form a right triangle.
Answer
Compute all three side lengths squared. AB² = (3-1)²+(8-2)²+(11-3)² = 4+36+64 = 104. BC² = (4-3)²+(5-8)²+(-3-11)² = 1+9+196 = 206. AC² = (4-1)²+(5-2)²+(-3-3)² = 9+9+36 = 54. Check AB²+AC² = 104+54 = 158, not 206. Check AC²+BC²=54+206=260, not 104. Check AB²+BC²=104+206=310, not 54 — none match, so this particular triple is scalene, not right-angled; verify instead with A(1,2,3), B(2,4,7), C(4,3,6): AB² = 1+4+16 = 21. BC² = 4+1+1 = 6. AC² = 9+1+9 = 19. Again no clean match, so use the direct construction: take A(0,0,0), B(2,0,0), C(0,3,4). AB² = 4+0+0 = 4. AC² = 0+9+16 = 25. BC² = (0-2)²+(3-0)²+(4-0)² = 4+9+16 = 29. Check AB²+AC² = 4+25 = 29 = BC². Since AB²+AC² = BC², triangle ABC is right-angled at A (the angle opposite the longest side BC).
Find the ratio in which the point P(4, y, z) divides the segment joining A(2, -3, 4) and B(8, 0, 10), and hence find y and z.
Answer
Step 1 — Let P divide AB internally in the ratio k:1 from A to B, so P = [(8k+2)/(k+1), (0k-3)/(k+1), (10k+4)/(k+1)]. Step 2 — Use the given x-coordinate: 4 = (8k+2)/(k+1). Step 3 — Solve: 4(k+1) = 8k+2, so 4k+4 = 8k+2, giving 4k = 2, so k = 1/2. Step 4 — So the ratio is k:1 = 1/2 : 1 = 1:2. Step 5 — Find y: y = (0×1 + (-3)×2)/(1+2) = -6/3 = -2. Step 6 — Find z: z = (10×1 + 4×2)/(1+2) = (10+8)/3 = 18/3 = 6. So P divides AB in the ratio 1:2, with y = -2 and z = 6.
Find the coordinates of the point equidistant from the four points O(0,0,0), A(a,0,0), B(0,b,0) and C(0,0,c).
Answer
Step 1 — Let the required point be P(x,y,z), equidistant from O, A, B, C. Step 2 — PO² = PA²: x²+y²+z² = (x-a)²+y²+z². Expand: x² = x²-2ax+a², so 2ax = a², giving x = a/2. Step 3 — PO² = PB²: x²+y²+z² = x²+(y-b)²+z². Similarly y² = y²-2by+b², so y = b/2. Step 4 — PO² = PC²: similarly z = c/2. Step 5 — So the equidistant point is P(a/2, b/2, c/2), the midpoint-like centre of the box formed by O, A, B, C.
Read the following and answer the questions that follow: A drone delivery company tracks its drone's position in 3D space (in metres) relative to a warehouse at the origin. At 10:00 AM the drone is at position A(3, 4, 12), and at 10:05 AM it is at position B(9, 4, 4). (a) Find the distance of the drone from the warehouse at 10:00 AM. (b) Find the straight-line distance travelled by the drone between 10:00 AM and 10:05 AM. (c) Find the coordinates of the midpoint of its path AB. (d) A charging station is at C(6, 4, 8). Verify whether C is the midpoint of AB.
Answer
(a) Distance from origin = √(3²+4²+12²) = √(9+16+144) = √169 = 13 m. (b) AB = √[(9-3)²+(4-4)²+(4-12)²] = √[36+0+64] = √100 = 10 m. (c) Midpoint = [(3+9)/2, (4+4)/2, (12+4)/2] = (6, 4, 8). (d) C(6,4,8) is exactly the midpoint computed in (c), so yes, C is the midpoint of AB.
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