Common Tangent to Parabolas: Exploring Mathematical Symmetry and Geometry

The intersection of mathematics and geometry often reveals fascinating insights into the properties of conic sections. One notable example involves finding the common tangent to the parabolas y^2 4x and x^2 4y. This exploration not only confirms the symmetry between these curves but also highlights the elegance of their mathematical properties.

Introduction to the Parabolas

The given parabolas y^2 4x and x^2 4y are symmetrically related to each other, as described. Specifically, one can be obtained from the other by a 90° rotation. This symmetry will be key in determining the common tangent line between the two parabolas.

Derivation of the Tangent Points and Tangent Line

To find the common tangent, we start by examining the slopes of the tangents at certain points on each parabola. For the parabola defined by y^2 4x, the slope of the tangent line can be found using implicit differentiation: 2yy' 4 y' frac{2}{y} For the second parabola, defined by x^2 4y, the slope of the tangent line is: 2x 4y' y' frac{x}{2} By symmetry, the common tangent must be parallel to the line y -x. Thus, the slope of the common tangent is -1. Let's find the points where this tangent line intersects the parabolas.

Contact Points and Equation of the Tangent Line

Let the tangent meet the first parabola at (x_0, y_0) and the second at (x_1, y_1). The slope of the tangent line to the first parabola is given by frac{y_0}{2}. Therefore, the equation of the tangent line is:

frac{y_0^2}{4} - x frac{y_0}{2} (y - y_0) y frac{2}{y_0} (x - frac{y_0^2}{2}) y_0

Since y_0 pm 2sqrt{x_0}, the equation simplifies to:

y sqrt{x_0} - frac{1}{sqrt{x_0}} x - 2sqrt{x_0}

Similarly, for the second parabola, the tangent line is:

y frac{x_1}{2} x - frac{x_1^2}{4}

Equating the slopes, we find:

frac{x_1}{2} -frac{1}{sqrt{x_0}} x_0 frac{4}{x_1^2}

Equating the intercepts, we get:

-frac{x_1^2}{4} sqrt{x_0} x_0 frac{x_1^4}{16}

From these equations, we find:

frac{4}{x_1^2} frac{x_1^4}{16} x_1^6 64 x_1 pm 2

Thus, x_0 1, y_0 pm 2, and x_1 -2, y_1 1. The equation of the common tangent is:

y -x - 1

Final Thoughts

The common tangent to the parabolas y^2 4x and x^2 4y is y -x - 1. This tangent line not only highlights the symmetry and geometric properties of the parabolas but also demonstrates the power of calculus in solving such problems.