Question

Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.b. At what point does the tangency occur?

Accepted Answer

Write down the equations of the two curves: the parabola $$y = p x$$^{2}$$ and the hyperbola x$$^{2}$$ - y$$^{2}$$ = 1 ($$considering only the right half, so $$x \geq 0). $$Express the hyperbola in terms of $$y to $$find $$y as a $$function of $$x$$: $$y = \pm \sqrt{x$$^{2}$$ - 1}. $$Since we consider the right half, $$x \geq 1$$ and $$y$$ can be either positive or negative depending on the branch. To find the point of tangency, set the $$y$$ values equal: $$p x$$^{2}$$ = \sqrt{x$$^{2}$$ - 1} ($$considering the positive branch for $$y). $$This gives a relation between $$x$$ and $$p at $$the tangency point. Find the derivatives (slopes) of both curves at the point of tangency. For the parabola, $$\frac{dy}{dx} = 2 p x. $$For the hyperbola, implicitly differentiate $$x^{2}$$ - $$y^{2} = 1$ to get 2x - 2y$$ \frac{dy}{dx} = 0$$, so $$\frac{dy}{dx} = \frac{x}{y}$$. Set the slopes equal at the tangency point: 2 p x$$ = \frac{x}{y}$$. Use the relation y = p$$ $$x^{2} to $$substitute for $$y$$ and solve for $$x. $$Then use this $$x to $$find the corresponding $$y$$ coordinate, which gives the point of tangency.

Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.
b. At what point does the tangency occur?

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Key Concepts

Equation of Tangent Lines to Curves

Parametric and Implicit Differentiation

Conditions for Tangency Between Two Curves