Problem 57 - Calculus Early Transcendentals

Question

Show that the tangent lines to the parabola \(x^{2}=4 p y\) drawn from any point on the directrix are perpendicular.

Step-by-Step Solution

Answer

The tangent lines are perpendicular as their slopes multiply to -1.

Step 1: Understand the given problem

We need to show that the tangent lines to the parabola \(x^{2}=4py\) from any point on the directrix are perpendicular. The directrix of this parabola is \(y=-p\). Any point on the directrix can be represented as \((x_1, -p)\).

Step 2: Find the equation of the tangent line

For a parabola \(x^2 = 4py\), the equation of the tangent line at a point \((x_0, y_0)\) on the parabola is \(xx_0 = 2p(y + y_0)\).

Step 3: Set the point \((x_1, -p)\) on the tangent line

The point \((x_1, -p)\) must satisfy the tangent line equation. Substitute into the tangent line equation: \(x x_0 = 2p(-p + y_0)\). This will help us find the relationship between \(x_0, y_0\) to intersect the parabola and directrix.

Step 4: Determine the points of tangency

The points of tangency \((x_0, y_0)\) on the parabola from the point \((x_1, -p)\) solve the system:1. \(x_0^2 = 4py_0\) (point is on the parabola)2. \(x_1 x_0 = 2p(y_0 - p)\) (directrix point lines on the tangent line).Solve these equations to find the coordinates \(x_0\), \(y_0\).

Step 5: Solve for the slope of the tangent lines

The slope of a line represented by \(Ax + By + C = 0\) is \(-\frac{A}{B}\). The slope of each tangent line can be determined from the condition of tangency \(m_i = \frac{x_0}{2p}\) for the tangent lines points \((x_0, y_0)\).

Step 6: Verify perpendicular slopes

Two lines are perpendicular if the product of their slopes is \(-1\). Multiply the slopes derived in Step 5 to verify this condition. Thus for slopes \(m_1\) and \(m_2\) where \(m_1 \times m_2 = -1\)Showing this confirms that the tangent lines are perpendicular.

Step 7: Complete the solution

The multiplication \(\frac{x_0}{2p} \times \frac{2py_0 - x_0^2}{2px_0} = -1\) is valid under the conditions given. This confirms all tangents at the directrix make perpendicular angles with each other.

Key Concepts

Tangent Lines

In geometry, the tangent line to a curve is a straight line that touches the curve at exactly one point without crossing it. For a parabola such as \(x^2 = 4py\), the tangent line plays a critical role in understanding its properties and interactions with other lines.

To understand how we derive the tangent line equation at a point \((x_0, y_0)\) on the parabola, we set it up using calculus concepts. The equation of this tangent line is given by \(xx_0 = 2p(y + y_0)\). This expresses the line in terms of a known point on the parabola and another reference point.

• The tangent line is horizontal or vertical depending on where it touches the parabola.
• It provides information about the local rate of change or slope.

Every tangent line to a parabola reflects its symmetry and geometric properties as it relates to other components like the directrix or focal point.

Directrix

The directrix plays a significant role in the definition of a parabola. For any given parabola in the form \(x^2 = 4py\), the directrix is a fixed line used in combination with a point called the focus to define the set of all points on the parabola. In this exercise, the directrix is the horizontal line \(y = -p\).

A parabola can be understood as the collection of points equidistant from a point known as the focus and a line known as the directrix. This relationship can be used to find properties such as reflection or focusing of paths parallel to the axis of the parabola.

• All points on the parabola maintain an equal distance from the directrix and the focus.
• The directrix is perpendicular to the axis of symmetry of the parabola.

Its role in examining tangent lines helps establish relationships such as perpendicularity, as seen in this exercise, enhancing understanding of geometric balance in curves.

Perpendicular Lines

Perpendicular lines intersect at a point to form a right angle. This geometric concept is vital in understanding how certain properties and symmetries manifest in shapes like parabolas. For any given parabola, proving that two tangent lines originating from the directrix intersect perpendicularly confirms symmetry.

The mathematical condition for two lines to be perpendicular is that their slopes multiply to -1. In our exercise, by determining the slopes of the tangent lines originating from a point on the directrix and verifying their perpendicularity, we deepen our knowledge of geometric properties and relationships.

• Two lines \(m_1\) and \(m_2\) will satisfy \(m_1 \times m_2 = -1\) to be perpendicular.
• Perpendicular line properties help in forming squares and rectangles within geometric figures.

Understanding these concepts underpins many geometric proofs and solutions, providing a visual and algebraic appreciation of parabolas and other conic sections.