- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
13.25. Miscellaneous review exercises on conic sections
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(a) Given $a ≠ 0.$ If the two parabolas $y^2 = 4p(x - a)$ and $x^2 = 4qy$ are tangent to each other, show that the $x$-coordinate of the point of contact is determined by $a$ alone.
(b) Find a condition on $a,$ $p,$ and $q$ which expresses the fact that the two parabolas are tangent to each other. -
(a) First, we express the parabola $y^2 = 4p(x - a)$ as a function of $x:$ $$y = 2\sqrt{p(x-a)}$$Then, we note that if the two parabolas are tangent, there exists some common point $(x_0, y_0)$ at which the derivatives of the two functions are equal. In other words, at $(x_0, y_0):$ $$\frac{x_0}{2q} = \sqrt{\frac{p}{x_0 - a}}$$Squaring both sides and rearranging terms gives us $$x_0^2(x_0 - a) = 4pq^2$$But we know that $(x_0 - a) = \frac{y_0^2}{4p}$ and that $y_0 = \frac{x_0^2}{4q},$ which means that $$(x_0 - a) = \frac{x_0^4}{64pq^2}$$And since $4pq^2 = x_0^2(x_0 - a),$ this simplifies to $$x_0^4 = 16x_0^2(x_0 - a)^2$$Solving for $x_0$ we find that $$x_0 = \frac{4}{3}a$$Thus proving that the $x$-coordinate of the point of contact is determined only by $a \quad \blacksquare$
(b) From (a) we know that the $x$-coordinate of the point of contact is $x_0 = \frac{4}{3}a,$ plugging this into the equations for the parabolas we get $$y_0^2 = \frac{4p}{3}a; \quad x_0^2 = 4qy_0 = \frac{16}{9}a^2$$And their respective derivatives at $(x_0, y_0)$ are $$\sqrt{\frac{p}{x_0 - a}}; \quad \frac{x_0}{2q}$$Setting $x_0 = \frac{4}{3}a$ and setting the derivatives equal to eachother, we get $$2a^2 = 9pq$$But since $x_0^2 = \frac{16}{9}a^2 = 4qy_0,$ this means that $a^2 = \frac{9}{4}qy_0$ and that $$2a^2 = 9pq = \frac{9}{2}qy_0$$meaning that $y_0 = 2p.$ And since $y_0^2 = \frac{4p}{3}a,$ this means that $a = 3p$ and $q = 2p.$
Plugging these values back into our previous equation of derivatives, we find the following condition on $a,$ $p,$ and $q$ which expresses the fact that the two parabolas are tangent to eachother: $$4a^3 = 27pq^2 \quad \blacksquare$$