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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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.

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주요 개념

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Equation of Tangency Between Curves

Tangency between two curves occurs when they touch at a point and share the same slope there. This means their equations must be satisfied simultaneously, and their derivatives (slopes) must be equal at the point of contact.

Parametric and Implicit Differentiation

To find slopes of curves defined implicitly or explicitly, differentiation techniques are used. For the hyperbola given implicitly, implicit differentiation helps find dy/dx, while for the parabola given explicitly, direct differentiation applies.

Substitution Method for Solving Systems

Solving for tangency involves substituting the parabola's equation into the hyperbola's equation to find common points. This reduces the problem to solving an equation in terms of p, enabling determination of the parameter for tangency.

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?