Problem 71 - Thomas Calculus

Question

The width of a parabola at the focus Show that the number 4\(p\) is the width of the parabola \(x^{2}=4 p y(p>0)\) at the focus by showing that the line \(y=p\) cuts the parabola at points that are 4\(p\) units apart.

Step-by-Step Solution

Answer

The width of the parabola at the focus is 4p.

Step 1: Identify the Focus

The given equation is \(x^2 = 4py\). This equation corresponds to a parabola with a vertical axis of symmetry. The vertex is at the origin \((0,0)\), and the focus is located at \((0, p)\).

Step 2: Substitute the Focus Line

To find the width of the parabola at the focus, substitute \(y = p\) into the parabola's equation \(x^2 = 4py\).

Step 3: Solve for x

Substituting \(y = p\), we get \(x^2 = 4p(p)\) which simplifies to \(x^2 = 4p^2\). Solving for \(x\), we take the square root of both sides: \(x = \pm 2p\).

Step 4: Calculate the Distance

The points where the line \(y = p\) intersects the parabola are \((2p, p)\) and \((-2p, p)\). The distance between these points is calculated as \((2p) - (-2p) = 4p\).

Key Concepts

Focus of a Parabola

The focus of a parabola is a crucial concept in understanding its geometric properties. For the equation of a parabola in the form of \(x^2 = 4py\), the focus is a point that lies on the axis of symmetry. In our example, the axis of symmetry is vertical since the parabola opens upwards or downwards. The vertex, which is often the point where the parabola intercepts the axis of symmetry, is at the origin \((0, 0)\) in our case. The focus of this parabola is at the point \((0, p)\). This simple point tells us a lot about the parabola:

• All points on the parabola are equidistant from the focus and a directrix, a line located opposite the vertex to the focus.
• It helps in constructing the parabolic shape by acting as a reference for the path of the curve.

The focus significantly influences the sound reflection properties and is essential in applications like satellite dishes and telescope mirrors. Remember, the closer the focus is to the vertex, the "tighter" or "narrower" the parabola appears.

Width of a Parabola

The width of a parabola at a particular point gives insights into how the curve behaves. Specifically, we look at the width of the parabola at the focus to define how "spread out" it is at that specific point. In our specific example, we explored a parabola described by \(x^2 = 4py\), and the line \(y = p\) intersects it.The width, in this case, was found to be \(4p\) using a straightforward calculation:

• Substitute \(y = p\) into the equation to establish the intersection points.
• This substitution leads to the equation \(x^2 = 4p^2\).
• Solving for \(x\) gives the points \(x = \pm 2p\).
• The distance between these points, \((2p, p)\) and \((-2p, p)\), confirms the width as \(4p\).

This width is not just a measurement but connects to the geometric properties of a parabola, influencing its light or sound reflection capabilities. It's a critical aspect of understanding where and how a parabola changes its curvature.

Equation of a Parabola

The equation of a parabola is fundamental in describing its shape and behavior. The standard form for a vertical parabola is \(x^2 = 4py\), where each part of the equation serves a specific purpose in detailing the parabola's properties:

• The \(x^2\) term part indicates symmetry along the \(y\)-axis, making the parabola open either up or down.
• The term \(4py\) defines how wide or narrow the parabola is and helps us locate the focus and the direction in which the parabola opens.

In the equation \(x^2 = 4py\), the vertex is always at the origin \((0, 0)\) unless altered by additional terms that shift the graph left, right, up, or down.Understanding the equation of a parabola allows us to quickly ascertain key features like:

• Vertex and Focus Location
• Direction and Width

This predictive power is why parabolas are immensely useful in various engineering and physics applications, such as signal representation, bridge design, and projectiles in motion.