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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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.

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주요 개념

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Equation of Tangent Lines to Curves

The tangent line to a curve at a point is the line that just touches the curve without crossing it locally. For a given curve, the slope of the tangent line is found by differentiating the curve's equation. Understanding how to find and express tangent lines is essential to determine where two curves share a common tangent.

Parametric and Implicit Differentiation

When dealing with curves defined implicitly or parametrically, implicit differentiation allows us to find the derivative dy/dx without explicitly solving for y. This technique is crucial for curves like hyperbolas, where y is not isolated, enabling the calculation of slopes needed for tangency conditions.

Conditions for Tangency Between Two Curves

Two curves are tangent at a point if they intersect there and their tangent lines have the same slope. This requires solving the system of equations given by the curves and equating their derivatives at the point of contact, ensuring both position and slope match.

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?