Normal to an Ellipse and Locus of Midpoints

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2015-03-21 22:50

This example demonstrates finding the normal line to a point on an ellipse, calculating its x and y intercepts, and then determining the locus of the midpoints of these intercepts. Key math concepts include ellipse equations, normal line equations, analytical geometry point-line relationships, midpoint formulas, and locus equations. GeoGebra commands used are Point, Tangent, OrthogonalLine, Intersect, and Midpoint. Suitable for high school analytical geometry instruction or math club exploration.

GeoGebra Commands & Parameters

Point(c) → A
Tangent(A, c) → a,
OrthogonalLine(A, a) → b
Intersect(b, yAxis) → B
Intersect(xAxis, b) → C
Midpoint(B, C) → D
Tangent(D, d) → e, g
OrthogonalLine(D, e) → f
Intersect(f, yAxis) → E
Intersect(xAxis, f) → F
Midpoint(E, F) → G
Tangent(G, h) → i,
OrthogonalLine(G, i) → j
Intersect(j, yAxis) → H
Intersect(j, xAxis) → I
Midpoint(H, I) → J
Tangent(J, k) → m,
OrthogonalLine(J, m) → n
Intersect(n, xAxis) → M
Intersect(n, yAxis) → N
Midpoint(M, N) → O