Textbook Question
53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
x² - y²/2 = 1
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16. Parametric Equations & Polar Coordinates
Conic Sections
3:48 minutesProblem 12.4.91
Textbook Question
90–94. Focal chords 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.
Let L be the latus rectum of the parabola y ² =4px for p>0. Let F be the focus of the parabola, P be any point on the parabola to the left of L, and D be the (shortest) distance between P and L. Show that for all P, D+|FP|+ is a constant. Find the constant.
Verified step by step guidance1
Identify the key elements of the problem: the parabola is given by the equation \(y^{2} = 4px\) with \(p > 0\), the focus \(F\) is at \((p, 0)\), and the latus rectum \(L\) is the vertical line through the focus, i.e., \(x = p\).
Express the coordinates of a general point \(P\) on the parabola to the left of \(L\). Since \(P\) lies on \(y^{2} = 4px\), its coordinates can be written as \(P = (x, y)\) where \(x = \frac{y^{2}}{4p}\) and \(x < p\) (to the left of \(L\)).
Calculate the shortest distance \(D\) from the point \(P\) to the latus rectum \(L\). Since \(L\) is the vertical line \(x = p\), the shortest distance is the horizontal distance: \(D = p - x\).
Find the distance \(|FP|\) between the focus \(F = (p, 0)\) and the point \(P = (x, y)\) using the distance formula: \(|FP| = \sqrt{(x - p)^{2} + y^{2}}\).
Set up the expression \(D + |FP|\) and simplify it by substituting \(x = \frac{y^{2}}{4p}\) and \(D = p - x\). Show that this sum simplifies to a constant independent of \(P\), and identify that constant.
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parabola and Its Focus
A parabola is the set of points equidistant from a fixed point called the focus and a fixed line called the directrix. For the parabola y² = 4px (p > 0), the focus is at (p, 0). Understanding the focus is essential because focal chords and distances from points on the parabola to the focus are central to the problem.
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Properties of Parabolas
Latus Rectum of a Parabola
The latus rectum is a special focal chord perpendicular to the axis of symmetry of the parabola. For y² = 4px, it passes through the focus and has endpoints on the parabola. Its length is 4p, and it helps define distances and geometric relationships involving points on the parabola.
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Distance Between Points and Lines
Calculating the shortest distance from a point to a line involves perpendicular projection. In this problem, the distance D from a point P on the parabola to the latus rectum line is crucial. Combining this with the distance |FP| from P to the focus helps establish the constant sum property.
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