Parabola Tangent to Two Lines
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This example demonstrates the geometric phenomenon of a parabola being tangent to two lines at two distinct points. Key math concepts include parabolas, tangents, coordinate geometry, and derivatives. Construction highlights involve using the ImplicitCurve command to plot implicit curves and objects like P, Q, A, B, C, D, S_1, S_2, a, b, f, g, h to build the geometric figure. Suitable for high school and above students to explore the properties and construction of tangent parabolas and lines.
GeoGebra Commands & Parameters
ImplicitCurve(((a * x) + (((-((A * (S_1 + S_2)) - (C * (S_1 + S_2)) - (2 * B) + (2 * D))) * a / ((((2 * A) * S_1) * S_2) - (((2 * C) * S_1) * S_2) - (B * (S_1 + S_2)) + (D * (S_1 + S_2)))) * y))^(2) + (((-2 * ((((2 * A^(3)) * S_1^(2)) * S_2^(2)) + (((2 * C^(3)) * S_1^(2)) * S_2^(2)) - (((2 * (((2 * S_1^(2)) * S_2) + (S_1 * S_2^(2)))) * C^(2)) * D) + (((2 * (S_1^(2) + ((2 * S_1) * S_2))) * C) * D^(2)) - ((2 * D^(3)) * S_1) - ((2 * B^(3)) * S_2) - ((2 * (((C * S_1^(2)) * S_2^(2)) - ((D * S_1) * S_2^(2)))) * A^(2)) + ((((2 * (((2 * S_1) * S_2) + S_2^(2))) * A) + ((S_1^(2) - ((2 * S_1) * S_2) - S_2^(2)) * C) - ((2 * D) * (S_1 - (2 * S_2)))) * B^(2)) - (((((2 * C^(2)) * S_1^(2)) * S_2^(2)) - ((((4 * C) * D) * S_1^(2)) * S_2) + ((S_1^(2) + ((2 * S_1) * S_2) - S_2^(2)) * D^(2))) * A) + (((((2 * C^(2)) * S_1^(2)) * S_2) - ((2 * ((S_1^(2) * S_2) + ((2 * S_1) * S_2^(2)))) * A^(2)) - ((((3 * S_1^(2)) + ((2 * S_1) * S_2) - S_2^(2)) * C) * D) + ((2 * D^(2)) * ((2 * S_1) - S_2)) + (((((4 * C) * S_1) * S_2^(2)) + ((S_1^(2) - ((2 * S_1) * S_2) - (3 * S_2^(2))) * D)) * A)) * B))) * a^(2) / ((((4 * A^(2)) * S_1^(2)) * S_2^(2)) + (((4 * C^(2)) * S_1^(2)) * S_2^(2)) + ((S_1^(2) + ((2 * S_1) * S_2) + S_2^(2)) * B^(2)) - (((4 * ((S_1^(2) * S_2) + (S_1 * S_2^(2)))) * C) * D) + ((S_1^(2) + ((2 * S_1) * S_2) + S_2^(2)) * D^(2)) - ((4 * ((((2 * C) * S_1^(2)) * S_2^(2)) - (((S_1^(2) * S_2) + (S_1 * S_2^(2))) * D))) * A) - ((2 * (((2 * ((S_1^(2) * S_2) + (S_1 * S_2^(2)))) * A) - ((2 * ((S_1^(2) * S_2) + (S_1 * S_2^(2)))) * C) + ((S_1^(2) + ((2 * S_1) * S_2) + S_2^(2)) * D))) * B))) * x) + (((2 * ((((2 * A^(3)) * S_1^(2)) * S_2) + (((2 * C^(3)) * S_1) * S_2^(2)) - (((2 * (((2 * S_1) * S_2) + S_2^(2))) * C^(2)) * D) + (((2 * C) * D^(2)) * (S_1 + (2 * S_2))) - ((((2 * (((2 * S_1^(2)) * S_2) - (S_1 * S_2^(2)))) * C) - ((S_1^(2) + ((2 * S_1) * S_2) - S_2^(2)) * D)) * A^(2)) + ((2 * ((A * ((2 * S_1) + S_2)) - (C * S_1) + D)) * B^(2)) - (2 * B^(3)) - (2 * D^(3)) + ((((2 * ((S_1^(2) * S_2) - ((2 * S_1) * S_2^(2)))) * C^(2)) - (((S_1^(2) - ((2 * S_1) * S_2) - (3 * S_2^(2))) * C) * D) - ((2 * D^(2)) * S_2)) * A) - ((((2 * (S_1^(2) + ((2 * S_1) * S_2))) * A^(2)) + ((S_1^(2) - ((2 * S_1) * S_2) - S_2^(2)) * C^(2)) + (((4 * C) * D) * S_2) - (((((3 * S_1^(2)) + ((2 * S_1) * S_2) - S_2^(2)) * C) - ((4 * D) * S_1)) * A) - (2 * D^(2))) * B))) * a^(2) / ((((4 * A^(2)) * S_1^(2)) * S_2^(2)) + (((4 * C^(2)) * S_1^(2)) * S_2^(2)) + ((S_1^(2) + ((2 * S_1) * S_2) + S_2^(2)) * B^(2)) - (((4 * ((S_1^(2) * S_2) + (S_1 * S_2^(2)))) * C) * D) + ((S_1^(2) + ((2 * S_1) * S_2) + S_2^(2)) * D^(2)) - ((4 * ((((2 * C) * S_1^(2)) * S_2^(2)) - (((S_1^(2) * S_2) + (S_1 * S_2^(2))) * D))) * A) - ((2 * (((2 * ((S_1^(2) * S_2) + (S_1 * S_2^(2)))) * A) - ((2 * ((S_1^(2) * S_2) + (S_1 * S_2^(2)))) * C) + ((S_1^(2) + ((2 * S_1) * S_2) + S_2^(2)) * D))) * B))) * y) + ((((4 * (((C * S_1^(2)) * S_2^(2)) - ((D * S_1^(2)) * S_2))) * A^(3)) - ((4 * ((C * S_2) - D)) * B^(3)) - (((((8 * C^(2)) * S_1^(2)) * S_2^(2)) - (((4 * (((3 * S_1^(2)) * S_2) + (S_1 * S_2^(2)))) * C) * D) + (((3 * S_1^(2)) + ((6 * S_1) * S_2) - S_2^(2)) * D^(2))) * A^(2)) + ((((S_1^(2) - ((6 * S_1) * S_2) - (3 * S_2^(2))) * C^(2)) + (((4 * C) * D) * (S_1 + (3 * S_2))) + ((4 * (((((2 * S_1) * S_2) + S_2^(2)) * C) - (D * ((2 * S_1) + S_2)))) * A) - (8 * D^(2))) * B^(2)) + ((4 * (((C^(3) * S_1^(2)) * S_2^(2)) - (((((2 * S_1^(2)) * S_2) + (S_1 * S_2^(2))) * C^(2)) * D) + (((S_1^(2) + ((2 * S_1) * S_2)) * C) * D^(2)) - (D^(3) * S_1))) * A) - ((2 * ((((2 * C^(3)) * S_1) * S_2^(2)) - (((2 * (((2 * S_1) * S_2) + S_2^(2))) * C^(2)) * D) + (((2 * C) * D^(2)) * (S_1 + (2 * S_2))) + ((2 * ((((S_1^(2) * S_2) + ((2 * S_1) * S_2^(2))) * C) - ((S_1^(2) + ((2 * S_1) * S_2)) * D))) * A^(2)) - (2 * D^(3)) - ((((2 * ((S_1^(2) * S_2) + ((3 * S_1) * S_2^(2)))) * C^(2)) - ((((3 * S_1^(2)) + ((10 * S_1) * S_2) + (3 * S_2^(2))) * C) * D) + ((2 * D^(2)) * ((3 * S_1) + S_2))) * A))) * B)) * a^(2) / ((((4 * A^(2)) * S_1^(2)) * S_2^(2)) + (((4 * C^(2)) * S_1^(2)) * S_2^(2)) + ((S_1^(2) + ((2 * S_1) * S_2) + S_2^(2)) * B^(2)) - (((4 * ((S_1^(2) * S_2) + (S_1 * S_2^(2)))) * C) * D) + ((S_1^(2) + ((2 * S_1) * S_2) + S_2^(2)) * D^(2)) - ((4 * ((((2 * C) * S_1^(2)) * S_2^(2)) - (((S_1^(2) * S_2) + (S_1 * S_2^(2))) * D))) * A) - ((2 * (((2 * ((S_1^(2) * S_2) + (S_1 * S_2^(2)))) * A) - ((2 * ((S_1^(2) * S_2) + (S_1 * S_2^(2)))) * C) + ((S_1^(2) + ((2 * S_1) * S_2) + S_2^(2)) * D))) * B)))) → b