Problem 90 - Calculus Early Transcendentals

Question

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The lines tangent to the endpoints of any focal chord of a parabola \(y^{2}=4 p x\) intersect on the directrix and are perpendicular.

Step-by-Step Solution

Answer

This verifies that the tangent lines are perpendicular, and we have proven both required properties: 1. The lines tangent to the endpoints of any focal chord of the parabola intersect on the directrix. 2. The tangent lines are perpendicular.

Step 1: Identify the Endpoints of the Focal Chord and the focus

The parabola equation is given as \(y^2=4px\). The focus of the parabola is \((p,0)\), and its directrix is the vertical line \(x=-p\). Let's denote the endpoints of the focal chord as \(A\) and \(B\). Since the focal chord is a line that passes through the focus and joins two points on the parabola, both points \(A\) and \(B\) must satisfy the equation \(y^2=4px\).

Step 2: Parameterize the Parabola Endpoints

To find an expression for the tangent lines, we'll first parameterize the coordinates of the points \(A\) and \(B\). We can do this by introducing a parameter \(t\), and writing the coordinates of \(A\) and \(B\) as \(A(at^2, 2at)\) and \(B(a/t^2,-2a/t)\), respectively, where \(a=2p\).

Step 3: Find the Equations of the Tangent Lines

Next, we'll find the equations of the tangent lines to the parabola at points \(A\) and \(B\). The slope of the tangent line at a given point on the parabola can be found by taking the derivative of the parabola equation with respect to \(x\), i.e., \(y'(x)=\frac{dy}{dx}=\frac{2}{t}\). Using the point-slope form of a line equation, we can write the equations of the tangent lines at points \(A\) and \(B\) as: Tangent at \(A: y-2at=\frac{2}{t}(x-at^2)\), and Tangent at \(B: y+2a/t=-\frac{t}{2}(x-a/t^2)\).

Step 4: Find the Intersection Point of the Tangent Lines

To find the point where the tangent lines intersect each other, set the right-hand sides of the tangent line equations equal to each other: \(\frac{2}{t}(x-at^2)=-\frac{t}{2}(x-a/t^2)\). Solving for \(x\), we obtain: \(x=\frac{a(t^4-1)}{t^4+1}\). The corresponding value of \(y\) can be found by substituting the value of \(x\) back into either tangent equation: \(y=2at\frac{t^4-1}{t^4+1}\).

Step 5: Show the Intersection Point is on the Directrix and the Tangent Lines are Perpendicular

Now, we must show that the intersection point of the tangent lines \((x,y)\) lies on the directrix \(x=-p\). Since \(a=2p\), the directrix can be written as \(x=-\frac{a}{2}\). Comparing the intersection point's x-coordinate, we find: \(x=\frac{a(t^4-1)}{t^4+1}=-\frac{a}{2}\). This shows that the intersection point indeed lies on the directrix. Next, we need to show that the tangent lines are perpendicular. To establish this, we can check if the product of their slopes is equal to -1: \(\frac{2}{t}\times -\frac{t}{2}=-1\).

Key Concepts

Conic Sections Properties

Conic sections are the curves formed when a plane intersects a double-napped cone at different angles. There are four primary types: circles, ellipses, parabolas, and hyperbolas. Each of these conic sections has unique properties. A parabola, the conic section involved here, is defined as the locus of points equidistant from a fixed point, called the focus, and a line called the directrix.

Parabolas have the characteristic property that any ray coming parallel to the axis of symmetry will reflect off the surface and pass through the focus. This reflective property is often harnessed in technologies like satellite dishes and flashlights. In the standard form, a parabola that opens rightwards can be described by the equation \(y^2=4px\), where \(p\) is the distance between the vertex of the parabola and the focus, and also between the vertex and the directrix.

Tangent Lines to Parabolas

Tangent lines to parabolas are straight lines that touch the parabola at exactly one point, without crossing the curve. The slope of a tangent line at any given point can be found using the derivative of the parabola's equation.

For a parabola described by \(y^2=4px\), the derivative, which represents the slope of the tangent, is \(y'(x)=\frac{dy}{dx}\). This derivative is crucial as it allows us to construct the equation of a tangent line at any point \((x_0, y_0)\) on the parabola using the point-slope formula for a line, \(y-y_0=m(x-x_0)\), where \(m\) is the slope provided by the derivative at that point.

Latus Rectum of Conics

The latus rectum of a conic section is a line segment perpendicular to the conic's axis of symmetry and passing through its focus. For parabolas, the latus rectum is incredibly significant since its length is directly related to the distance between the focus and the directrix.

In the parabola \(y^2=4px\), the latus rectum has a length of \(4p\). This property is helpful in many geometrical problems and applications where the focus is a point of interest, such as in the design of reflective optics and during the analytical study of projectile motions.

Parameterization of Parabola Endpoints

Parameterizing the endpoints of a focal chord involves expressing the coordinates of these points in terms of a single variable. This method is useful when dealing with problems that involve constructing tangents or normals to parabolas at specific points.

In our problem, we parameterize endpoints \(A\) and \(B\) using a parameter \(t\), leading to the coordinates \(A(at^2, 2at)\) and \(B(a/t^2, -2a/t)\), where \(a=2p\). This technique not only simplifies the calculation but also links the geometrical properties directly to the algebraic parameter \(t\), making it easier to manipulate and understand their relations.

Derivative Slope of Parabola

The slope of a tangent line to a parabola at any point is equal to the derivative of the parabola's equation at that point. For a parabola given by \(y^2=4px\), we differentiate with respect to \(x\) to obtain the slope of the tangent line.

We find that \(y'(x)=\frac{dy}{dx}=2p/y\), which, with the parameterization method, translates to \(y'(x)=\frac{2}{t}\) when at point \(A\) and \(-\frac{t}{2}\) at point \(B\). The product of the slopes of the two tangent lines is \(-1\), confirming they are perpendicular. Understanding the relationship between the derivative and the slope of the tangent lines at various points is fundamental in calculus particularly when analyzing the behavior and properties of parabolic curves.