Question

Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.a. For what value of p is P tangent to H?

Accepted Answer

Start by writing down the equations of the two curves: the parabola $$P is $$given by $$y = p x$$^{2}$$, and the hyperbola H$ is given by x$$^{2}$$ - y$$^{2}$$ = 1 ($$considering only the right half, so $$x \geq 0). $$Substitute the expression for $$y$$ from the parabola into the hyperbola equation to find the points of intersection. This gives: $$x^{2} - (p$$ $$x^{2}$$)^{2}$$ = 1$$, which simplifies to $$x^{2}$$ - $$p^{2}$$ $$x^{4} = 1$. Rewrite the equation as a polynomial in x$$^{2}$$: p$$^{2}$$ x$$^{4}$$ - x$$^{2}$$ + 1 = 0. $$Let $$z = x$$^{2}$$ to get p$$^{2}$$ z$$^{2}$$ - z + 1 = 0. $$For the parabola and hyperbola to be tangent, the equation must have exactly one solution for $$z$$, meaning the quadratic in $$z$$ has a single root. Set the discriminant of this quadratic to zero: $$\Delta = (-1$$)^{2}$$ - 4 \cdot p$$^{2}$$ \cdot 1 = 0. $$Solve the discriminant equation $$1 - 4 p$$^{2}$$ = 0$$ for $$p to $$find the value(s) of $$p$ where the parabola is tangent to the hyperbola.

Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.
a. For what value of p is P tangent to H?

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Key Concepts

Equation of Tangency Between Curves

Parametric and Implicit Differentiation

Substitution Method for Solving Systems