91Ó°ÊÓ

A family of curves refers to a set of curves that share a common mathematical relationship or equation but differ when a certain parameter within that equation is varied. In our problem, the family of curves is given by the equation \(x + y = cx^2\), where \(c\) is a parameter that can take on different values.
This means each value of \(c\) produces a different curve, but all curves will have the same basic form. You can imagine a family of curves as contours on a topography map, where each contour (curve) represents a different level (value of \(c\)).
Finding a family of curves is useful because it allows us to study how shifting one aspect of a system (in this case, the parameter \(c\)) affects the entire set of possible outcomes, thus offering a deeper understanding of the relationship between \(x\) and \(y\) for varying \(c\).
This shared structural form is how we begin to explore their intersecting trajectories, especially when these intersections happen at a specific angle.

The derivative is a fundamental concept in calculus that represents the rate of change of a function. Derivatives indicate how a function's output value changes as its input changes.
For our exercise, we start by finding the derivative of the curve equation \(x + y = cx^2\). This derivative \(\frac{dy}{dx}\) will help us understand how steep the curve is at any point, which is crucial for figuring out how to intersect it at an angle.
Calculating the derivative, we differentiate with respect to \(x\), resulting in \(\frac{dy}{dx} = y' = -1 + 2cx\). This means that at any point \((x, y)\) on the curve, the slope is \(-1 + 2cx\).
Knowing how to calculate and interpret derivatives is key to exploring more complex properties of functions, like finding where two functions might intersect at a given angle.

The tangent angle formula is used to determine the angle between two curves at the point where they intersect. It's a powerful tool in this problem as we're looking for trajectories intersecting at a specific angle.
When two curves intersect, their corresponding derivatives at the intersection point indicate their steepness. The tangent angle formula is
\[\frac{y' - f'(x)}{1 + y' f'(x)} = m\]
where \(y'\) and \(f'(x)\) are the derivatives of the intersecting curves, and \(m\) is the tangent of the angle between them.
In this scenario, we have \(\tan a = 2\) for the original intersection angle, and hence we use the negative reciprocal for the desired intersection which becomes \(-2\). Plugging this value into the tangent angle formula allows us to find the derivative of the second curve that intersects our given family of curves at this specific angle.

Integration is essentially the reverse process of differentiation. While derivatives concern themselves with rates of change, integration is about accumulation and finding areas under curves.
After finding \(f'(x) = \frac{1}{3(2cx - 1)}\), the next step is to integrate this derivative to obtain the family of trajectories that intersect our initial family of curves.
Integrating \(\frac{1}{3(2cx - 1)}\) with respect to \(x\) results in the antiderivative \(y_2(x) = \int \frac{dx}{3(2cx - 1)} + C\), where \(C\) is an arbitrary constant representing vertical shifts of the trajectories. This integral represents the new family of curves—our "oblique trajectories."
Integration often involves finding these antiderivatives and setting up initial conditions or boundaries described by additional parameters or constants like \(C\) to fully define a particular solution within the family.