Problem 66 - Precalculus with Calculus Previews

Question

Tangent Lines In this problem it is necessary that you have studied Section 2.10 . (a) Any two distinct tangent lines to a parabola must intersect. Show that the tangent lines to a parabola at the endpoints of its focal chord are perpendicular. [Hint: Without loss of generality we may assume that an equation of the parabola is given by \(x^{2}=4 c y\).] (b) Show that the tangent lines at the endpoints of the focal chord of a parabola intersect on its directrix.

Step-by-Step Solution

Answer

The tangent lines at focal chord endpoints of a parabola are perpendicular and intersect on the directrix.

Step 1: Understand the equation of the parabola

Given the parabola equation is \(x^2 = 4cy\), which has its vertex at the origin (0,0) and its focus at (0,c). The equation represents a parabola opening upwards.

Step 2: Determine focal chord endpoints

Endpoints of the focal chord are symmetrical. Let \((x_1, y_1)\) and \((x_2, y_2)\) represent the endpoints; utilize the relation that \(y = \frac{x^2}{4c}\) when applying it to points on the parabola satisfying \(x_1x_2 = c^2\). This focal chord condition holds for two points on the parabola equidistant from its vertex.

Step 3: Derive tangent line equations at endpoints

The derivative of \(x^2 = 4cy\) is \(2x = 4c \frac{dy}{dx}\), leading to \(\frac{dy}{dx} = \frac{x}{2c}\). For endpoint \((x_1, y_1)\), the slope is \(m_1 = \frac{x_1}{2c}\), and the tangent line is \(y - y_1 = m_1(x - x_1)\). Similarly, compute the tangent for point \((x_2, y_2)\).

Step 4: Show angles between tangent lines are perpendicular

Verify perpendicularity in tangent lines by evaluating the product of their slopes: \(m_1 \cdot m_2 = \frac{x_1}{2c} \cdot \frac{x_2}{2c} = \frac{x_1 x_2}{4c^2}\). Given \(x_1 x_2 = c^2\), the product becomes \(1\), concluding the slopes are opposite reciprocals, confirming perpendicularity.

Step 5: Show intersection on the directrix

The directrix of the parabola \(x^2 = 4cy\) is \(y = -c\). Solve the system of equations of the tangent lines (from Step 3). Their intersection point will yield \((x_0, y_0)\) such that substituting \(y_0 = -c\) resolves the intersection point on the directrix.

Key Concepts

Parabola

A parabola is a symmetrical, U-shaped curve that is defined by a quadratic equation. In this context, we are dealing with the parabola given by the equation \(x^2 = 4cy\). This equation represents a parabola that opens upwards, with its vertex located at the origin \((0,0)\) and its focus at the point \((0, c)\). The distance \(c\) measures how "open" or "narrow" the parabola appears. In general, the properties of a parabola include:

• A single axis of symmetry, which is the y-axis for our given parabola.
• A vertex, which acts as a central "point" for the parabola.
• A focus, which helps define the shape of the parabola.
• A directrix, a line used to ensure every point on the parabola is equidistant from the focus and this line.

Understanding these elements helps in determining points such as the focal chord and intersections of lines associated with the parabola.

Focal Chord

A focal chord of a parabola is a line segment that passes through the focus of the parabola and has its endpoints on the parabola itself. In our problem, we consider a focal chord within the parabola \(x^2 = 4cy\). These chords have a special property because the endpoints are symmetrical with respect to the axis of symmetry of the parabola.

To find the endpoints of the focal chord, assume them to be \((x_1, y_1)\) and \((x_2, y_2)\). For a parabola of the form \(x^2 = 4cy\), the condition \(x_1 x_2 = c^2\) holds true. This symmetrical property is vital in proving the perpendicular nature of tangents at these endpoints and their intersection at a specific point (the directrix), linking geometric and algebraic concepts.

Directrix

The directrix is a fixed line used in conjunction with the focus to define a parabola. For the parabola \(x^2 = 4cy\), the directrix is the horizontal line \(y = -c\).

This line is crucial because each point on the parabola is equidistant from the directrix and the focus. This distance feature plays an instrumental role when working with tangents to the parabola. In part (b) of the problem, the intersection of the tangent lines at the endpoints of the focal chord with the directrix is shown. Knowing the directrix helps us locate this intersection point by solving the system of equations generated from tangent lines, confirming they meet at \(y = -c\).

Perpendicular Slopes

When two lines are perpendicular, their slopes are negative reciprocals of each other. In the context of this problem, the tangents at the endpoints of the focal chord of a parabola are perpendicular. For the given parabola, the slope of the tangent at any point \((x, y)\) is \(\frac{x}{2c}\).

Given endpoints \((x_1, y_1)\) and \((x_2, y_2)\) of the focal chord:

• The slope of the tangent at \((x_1, y_1)\) is \(m_1 = \frac{x_1}{2c}\).
• The slope of the tangent at \((x_2, y_2)\) is \(m_2 = \frac{x_2}{2c}\).

This results in their product being \(m_1 \cdot m_2 = \frac{x_1 x_2}{4c^2}\), which simplifies to \(1\) using \(x_1 x_2 = c^2\). As such, the tangents are confirmed perpendicular, allowing us to easily analyze their intersection behavior.