Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. The hyperbola y²/2 - x²/4 = 1 has no x-intercept.
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16. Parametric Equations & Polar Coordinates
Conic Sections
2:00 minutesProblem 12.4.83a
Textbook Question
Reflection property of parabolas: Consider the parabola y = x²/(4p) with its focus at F(0, p). The goal is to show that the angle of incidence (α) equals the angle of reflection (β).
a. Let P(x₀, y₀) be a point on the parabola. Show that the slope of the tangent line at P is tan θ = x₀/(2p).
Verified step by step guidance1
Start with the given parabola equation: \(y = \frac{x^2}{4p}\). To find the slope of the tangent line at a point \(P(x_0, y_0)\) on the parabola, we need to compute the derivative \(\frac{dy}{dx}\).
Differentiate \(y = \frac{x^2}{4p}\) with respect to \(x\) using the power rule: \(\frac{dy}{dx} = \frac{2x}{4p} = \frac{x}{2p}\).
Evaluate the derivative at the point \(x = x_0\) to find the slope of the tangent line at \(P\): \(\text{slope} = \frac{x_0}{2p}\).
Recall that the slope of the tangent line is the tangent of the angle \(\theta\) that the tangent line makes with the positive \(x\)-axis. Therefore, \(\tan \theta = \frac{x_0}{2p}\).
This completes the proof that the slope of the tangent line at \(P\) on the parabola \(y = \frac{x^2}{4p}\) is \(\tan \theta = \frac{x_0}{2p}\).
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative and Slope of Tangent Line
The derivative of a function at a point gives the slope of the tangent line to the curve at that point. For the parabola y = x²/(4p), differentiating with respect to x yields dy/dx = x/(2p), which represents the slope of the tangent line at any point (x₀, y₀). This slope is crucial for understanding the angle θ the tangent line makes with the x-axis.
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Reflection Property of Parabolas
A parabola reflects rays coming parallel to its axis of symmetry through its focus. The reflection property states that the angle of incidence (α) equals the angle of reflection (β) at any point on the parabola. This property is fundamental in optics and is derived using the geometry of the parabola and the tangent line at the point of incidence.
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Angle Relationships and Trigonometry in Geometry
Understanding the angles formed by the tangent line, the line from the focus to the point on the parabola, and the horizontal axis involves trigonometric relationships. The angles α, β, θ, and φ relate through tangent and reflection properties, allowing the use of slope (tan θ) and geometric reasoning to prove angle equality.
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