Locus of Intersections of Normal Lines with the xy-Plane
Multivariable calculus
core
Let normal lines be drawn at all points on the surface $z = ax^2 + by^2$ that are at a given height $h$ above the xy-plane. Find an equation of the curve in which these lines intersect the xy-plane.
multivariable-calculusnormal-linegradientlevel-surfaceparametric-equationsintersection-of-line-and-planelocus-of-pointspartial-derivativessurfacesfind-equationeliminate-parameterthree-dimensional-geometry
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What you need first
- how to find the gradient of a multivariable function
- how to write parametric equations for a line in 3D space
- how to eliminate parameters to find a Cartesian equation
What this problem tests
Tests the student's ability to find normal vectors using gradients, construct parametric line equations, and eliminate variables to find a geometric locus.
This problem is typical of a multivariable calculus course, testing the ability to combine gradients, lines in 3D space, and algebraic manipulation to find a locus of points.
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