Textbook Question
53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
4x² + 8y² = 16
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16. Parametric Equations & Polar Coordinates
Conic Sections
5:45 minutesProblem 12.R.76
Textbook Question
General equations for a circle Prove that the equations
X = a cos t + b sin t, y = c cos t + d sin t
where a, b, c, and d are real numbers, describe a circle of radius R provided a² +c² =b² +d² = R² and ab+cd=0.
Verified step by step guidance1
Start with the given parametric equations of the curve: \(X = a \cos t + b \sin t\) and \(y = c \cos t + d \sin t\), where \(a, b, c,\) and \(d\) are real numbers.
Square both equations and add them together to find an expression for \(X^2 + y^2\):
\(X^2 + y^2 = (a \cos t + b \sin t)^2 + (c \cos t + d \sin t)^2\).
Expand the squares using the distributive property:
\(X^2 + y^2 = (a^2 \cos^2 t + 2ab \cos t \sin t + b^2 \sin^2 t) + (c^2 \cos^2 t + 2cd \cos t \sin t + d^2 \sin^2 t)\).
Group like terms involving \(\cos^2 t\), \(\sin^2 t\), and \(\cos t \sin t\):
\(X^2 + y^2 = (a^2 + c^2) \cos^2 t + (b^2 + d^2) \sin^2 t + 2(ab + cd) \cos t \sin t\).
Use the given conditions \(a^2 + c^2 = b^2 + d^2 = R^2\) and \(ab + cd = 0\) to simplify the expression:
\(X^2 + y^2 = R^2 \cos^2 t + R^2 \sin^2 t + 0 = R^2 (\cos^2 t + \sin^2 t)\).
Since \(\cos^2 t + \sin^2 t = 1\), this reduces to \(X^2 + y^2 = R^2\), which is the equation of a circle with radius \(R\).
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Video duration:
5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parametric Equations of a Curve
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. In this problem, x and y are given in terms of trigonometric functions of t, which allows describing complex curves like circles by varying t over an interval.
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Parameterizing Equations
Properties of the Circle Equation
A circle with radius R centered at the origin satisfies x² + y² = R². To prove a parametric form represents a circle, one must show that substituting x(t) and y(t) into this equation yields a constant radius R, independent of t.
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Trigonometric Identities and Orthogonality Conditions
The conditions a² + c² = b² + d² = R² and ab + cd = 0 ensure orthogonality and equal magnitude of vectors formed by coefficients. These constraints use trigonometric identities to guarantee that the parametric equations trace a circle rather than an ellipse or other shape.
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