Find the angle at which the parabolas y² = 4x + 4 and y² = -6x + 9 intersect at each of their intersection points. A) 90º at each intersection point B) 120º at each intersection point C) 60º at each intersection point D) 75º at each intersection point E) 50º at each intersection point

Question

Quizplus · Accepted Answer

The slopes of the tangents to the parabolas at the points of intersection represent the direction of the parabolas at those points. The slope of the tangent to a parabola $y = ax^2 + bx + c$ at any point is given by the derivative $dy/dx = 2ax + b$. For the given parabolas $y = 4x + 4$ and $y = -6x + 9$, the slopes are 4 and -6, respectively. The angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by $\tan(\theta) = |(m_2 - m_1) / (1 + m_1m_2)|$. Substituting $m_1 = 4$ and $m_2 = -6$, we find $\tan(\theta) = |(-6 - 4) / (1 + 4(-6))| = |(-10) / (1 - 24)| = 10/23$. However, this calculation is incorrect for determining the angle between the parabolas; the correct approach is to use the fact that the product of the slopes of two perpendicular lines is -1. Since the slopes of the tangents to the parabolas are 4 and -6, their product is $4 \times -6 = -24$, which does not satisfy the condition for perpendicularity. The correct approach to find the angle between the parabolas involves recognizing that if the slopes were perpendicular, the angle between them would be 90 degrees. Given the misunderstanding in the calculation, the correct explanation is that when two lines are perpendicular, the angle between them is indeed 90 degrees, but the calculation provided does not correctly apply to finding the angle between the parabolas. The correct answer is based on the principle that if the tangents to the curves at the intersection points were perpendicular, the angle between the curves would be 90 degrees. However, the explanation mistakenly attempted to calculate the angle directly from the slopes, which is not how the angle between intersecting curves is determined in this context. The correct answer is that the angle at which the parabolas intersect cannot be determined with the information and approach provided, but the principle that leads to the selection of 90 degrees is based on a misunderstanding of how to apply the slopes of the tangents in this context.

Find the Angle at Which the Parabolas Y2 = 4x

Find the Angle at Which the Parabolas Y² = 4x