📝 Translated quadric surfaces (26 MCQs)

📖 Explanation: The student applied the opposite signs in the substitution. Translating by $\langle h,k,l \rangle$ requires replacing $x$ with $x-h$, not $x+h$. This is a common sign-reversal misconception when students memorize formulas without understanding that the new coordinates must satisfy the original equation relative to the shifted origin.

📖 Explanation: Translation preserves all intrinsic metric properties including shape, size, and orientation. The asymptotic cone’s orientation and focal distances are invariant because they depend only on coefficients, not position. Vertices change location but maintain relative positioning. Understanding invariance distinguishes rigid motions from deformations in quadric surface analysis.

📖 Explanation: The narrowest cross-section of a hyperboloid of one sheet occurs at its center where $y=4$. Moving this to $y=0$ requires subtracting 4 from y-coordinates, meaning we add 4 to the variable: replace $y$ with $y-4$ becomes $y$, requiring translation vector $\langle 0,4,0 \rangle$. This tests application of geometric features to real-world repositioning.

📖 Explanation: Comparing equations shows P2 results from substituting $x \to x-1$, $y \to y+2$, $z \to z-3$ in P1, confirming pure translation by $\langle 1,-2,3 \rangle$. Translation preserves all differential properties including Gaussian curvature and focal parameters. Students often mistakenly believe position affects shape; this question targets that conceptual gap through direct comparison.

📖 Explanation: Dividing by 36 gives $\frac{(x-1)^2}{9} + \frac{(y+2)^2}{4} + \frac{z^2}{36} = 1$. Translation doesn’t alter axis lengths; the y-semi-axis is $\sqrt{4} = 2$. This direct recall question verifies foundational knowledge that translation affects only position terms, not denominators representing squared semi-axes. Essential baseline for higher-order problems involving transformed quadrics.

📖 Explanation: Quadric classification is determined by the signature of the quadratic form (signs of eigenvalues), unaffected by linear translation terms. Changing $a,b,c$ merely relocates the center while maintaining the same canonical form. Options A and B express this invariant principle differently; recognizing their equivalence demonstrates deep conceptual understanding beyond rote memorization of classification rules.

📖 Explanation: Standard paraboloid $z = \frac{1}{4p}(x^2+y^2)$ has focus at $(0,0,p)$. Here $p=2$, so unshifted focus is at $z=2$. Receiver at $z=12$ implies vertical shift of 10 units. Thus translation vector is $\langle 0,0,10 \rangle$. This multi-step problem combines focal property knowledge with translation application in engineering context, testing integrated modeling skills.

📖 Explanation: Contour centers directly reflect xy-translation since level curves $z=k$ yield $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = k-c$, centered at $(h,k)$. The z-value where contours begin indicates vertical translation. This graph-based interpretation connects visual pattern recognition to algebraic parameters, assessing ability to extract translation information from projected views rather than explicit equations.

📖 Explanation: Direct substitution maintains structural clarity, allowing instant validation that each variable shift matches intended translation. Expansion risks computational mistakes and obscures parameter correspondence. However, completing squares serves as independent verification. Recognizing complementary strengths of methods reflects metacognitive awareness of mathematical practice, going beyond procedural execution to strategic tool selection based on purpose.

📖 Explanation: The saddle point of $z=xy$ is at origin. Substituting $X=x-1$, $Y=y+3$, $Z=z-2$ transforms to $Z=XY$, whose saddle is at $X=Y=Z=0$. Solving gives $x=1, y=-3, z=2$. This requires understanding that saddle points translate like centers despite lacking symmetry axes, testing nuanced grasp of critical point behavior under affine transformations.

📖 Explanation: Ellipsoid extends ±5 in x-direction from center x=10, so minimum x-coordinate is 10−5=5. Clearance from x=0 plane is 5 units, exceeding 3-unit requirement. This scenario-based problem integrates geometric extent calculation with regulatory constraint checking, demanding careful distinction between center position and surface boundary in applied contexts. Misconception about measuring from center versus surface is targeted in distractors.

📖 Explanation: Original normalized form is $\frac{x^2}{9} + \frac{y^2}{36} + \frac{z^2}{4} = 1$. Target has same denominators but permuted variables and shifted center. Scaling isn't needed since semi-axes already match target values after normalization. Only translation by $\langle 2,-1,3 \rangle$ and relabeling axes achieves result. However, if interpreted as literal variable mapping, scaling precedes translation in standard reduction procedure. This mixed-concept question tests understanding of canonical form derivation order.

📖 Explanation: Gaussian curvature is an intrinsic property preserved under isometries including translations. For $z=x^2-y^2$, $K = \frac{-4}{(1+4x^2+4y^2)^2}$; after translation, expression shifts arguments but functional form and values at corresponding points remain identical. This challenges advanced misconception linking extrinsic position to intrinsic geometry, requiring differential geometry insight beyond basic quadric classification.

📖 Explanation: Contour center gives xy-translation $(2,-3)$. Density gradient indicates extremum type but not absolute z-position. Without knowing $z$-coordinate at center, vertical translation remains undetermined. Axis orientation and opening direction affect shape classification but not translation parameters once center xy is known. This graph-based inference question highlights insufficiency of partial visual data and necessity of combining multiple cues for complete parameter recovery.

📖 Explanation: Nonzero residual mean suggests constant offset, but correlation with $x,y,z$ specifically reveals linear trends indicating unmodeled translation. If residuals increase with $x$, likely $x$-translation missing. This targeted diagnostic leverages statistical patterns to pinpoint specific parameter omission, demonstrating applied error analysis skill beyond generic goodness-of-fit assessment. Correlation structure encodes directional information absent in aggregate statistics.

📖 Explanation: Set $z=0$: $\frac{(x-4)^2}{16} + \frac{(y+3)^2}{9} = 1 - \frac{100}{25} = -3$. Negative right side means no real solution; dome minimum z is $10-5=5$. But wait—recalculating: $\frac{(z-10)^2}{25} \leq 1$ implies $z \in [5,15]$. At $z=0$, LHS >1 always. Actually, no intersection exists. Correction: Option A is correct. This trick question tests careful domain analysis before computation, exposing tendency to mechanically solve without checking feasibility—a critical HOTS in modeling validation.

📖 Explanation: All quadrics related by translation are geometrically congruent; classification depends exclusively on quadratic form signature. Translation creates positional variants within equivalence classes defined by invariants. This conceptual synthesis integrates algebraic, geometric, and categorical perspectives, forming foundational understanding for advanced topics like moduli spaces. Distractors represent common overgeneralizations or underappreciations of translation’s limited but precise role.

📖 Explanation: To center at origin, apply inverse translation: replace $x \to x+3$, etc., corresponding to moving surface by $\langle -3,1,-5 \rangle$. This eliminates linear terms in shader code. Option D misorders operations; normalization doesn’t affect center. Graphics optimization context makes abstract translation concrete, testing transfer of mathematical concept to computational implementation where performance depends on canonical positioning.

📖 Explanation: Gradient $\nabla V = \langle 2(x-2), 8(y+1), 18(z-3) \rangle$ vanishes immediately at $(2,-1,3)$. Expansion yields coupled linear system requiring solving. Translation reveals equilibrium as center by construction. Hessian is already diagonal regardless, but translation makes critical point obvious. This mixed-concept question links multivariable calculus to geometric insight, showing how structural recognition supersedes brute-force computation in identifying physical equilibria.

📖 Explanation: Translated surface: $(x-a)^2 + (y-b)^2 - (z-c)^2 = 1$. Line satisfies iff $(t-a)^2 + (t-b)^2 - (t-c)^2 = 1$ for all t. Expanding: $t^2(1+1-1) + t(-2a-2b+2c) + (a^2+b^2-c^2-1) = 0$. Coefficient of $t^2$ is 1≠0, impossible unless... Wait, recalculate: coefficient is 1, so identity requires 1=0 contradiction. But if line lies on surface, equation must hold identically. Since leading coefficient ≠0, no solution exists. However, reconsider: maybe specific t values? No, “containing the line” means all t. Thus D seems correct. But let’s verify special case: if a=b=c, then expression becomes $t^2 - 2at + (2a^2 - a^2 -1) = t^2 -2at + (a^2-1)$, still quadratic. Indeed, no translation works because hyperboloid contains no straight lines through arbitrary directions? Actually hyperboloid of one sheet contains two families of lines, but not necessarily this diagonal. Final answer: D. This Olympiad-style problem demands rigorous identity analysis and knowledge of ruled surface properties, pushing beyond standard curriculum.

📖 Explanation: Centered quadrics exploit symmetry for error cancellation in quadrature rules. Coordinate alignment minimizes discretization artifacts when grids are axis-aligned. Well-conditioned parametrizations near origin improve numerical stability. These combined effects make canonical positioning advantageous computationally. This advanced application question bridges theoretical geometry with numerical analysis, illustrating how mathematical structure influences algorithmic performance beyond mere representation convenience.

📖 Explanation: Linear terms indicate off-center position; classification requires canonical form achieved by completing squares to isolate translation. Only after removing D,E,F via translation can signature analysis correctly identify surface type. Rotation handles cross-terms, not linear ones. This procedural sequencing question tests understanding that translation reduction is prerequisite to classification, addressing common student error of attempting type identification on non-canonical forms.

📖 Explanation: Minkowski sum of ellipsoid and sphere yields larger concentric ellipsoid with semi-axes incremented by sphere radius. Translation components unchanged since sphere is symmetric. New semi-axes: $2+0.5=2.5$, $3+0.5=3.5$, $1+0.5=1.5$. This spatial reasoning problem applies translation concepts to collision avoidance, requiring understanding that uniform dilation preserves center while modifying scale parameters—a non-obvious extension of basic translation knowledge to geometric operations.

📖 Explanation: Equation equals zero only at single point (1,2,3); it’s degenerate, not a sphere. Standard sphere requires positive RHS equal to radius squared. This error-analysis question targets confusion between limiting cases and proper definitions, emphasizing that translation alone cannot create non-degenerate objects from degenerate ones. Recognizing degeneracy is crucial for avoiding invalid geometric interpretations in proofs and applications.

📖 Explanation: Center is at $(t,0,0)$ for each t, tracing x-axis. Translation parameter t directly controls center x-coordinate while y,z centers remain fixed. This dynamic perspective question shifts focus from static surfaces to parametric families, testing understanding of translation as continuous deformation generator. Locus characterization connects individual surface properties to collective behavior, fostering higher-level abstraction skills essential for advanced geometry.

📖 Explanation: Automated validation should parse equation into canonical form via completing squares, extract actual center $(-1,2,-3)$, and compare to user-declared (1,2,3). Sign mismatch in linear terms causes center inversion. This quality-control scenario emphasizes programmatic verification of human input, translating mathematical consistency checks into software logic. It reinforces that translation parameters must align precisely between symbolic and metadata representations in technical workflows.