Envelopes
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Envelopes - Exercise 1A - Problems in finding envelopes
Questions 1.Find the envelope of the family of circles [Equation goes here - download the original to see it.] 2. Find the envelope of a family of circles of the same radius R whose centres lie on the x-axis. 3. Find the envelope of a family of straight lines that form with the coordinate axes a triangle of constant area S. 4. Find the envelope of the family of circumferences with centres on an ellipse and passing through one of its foci. 5. Find the envelope of the family of ellipses that have a given area and common principal axes. 6. Find the envelope of the family of ellipses that have common principal axes and a given semi-axis sum. 7. Find the envelope of the family of circumferences constructed on parallel chords of a circumferences as on diameter. 8. Show that the envelope of the family of circles which pass through the origin and have centres on the circle [Equation goes here - download the original to see it.] is the cardioids with polar equation [Equation goes here - download the original to see it.] 9.Find the envelope of the rays reflected from a circumference if the luminous point is on the circumference. 10. Sketch the family of circles which have centres on [Equation goes here - download the original to see it.] and touch the x-axis. Show geometrically that the envelope is a pair of straight lines. Find these lines: (i) by using [Equation goes here - download the original to see it.] , where [Equation goes here - download the original to see it.] (ii) by using the equation of the circles with centre [Equation goes here - download the original to see it.] and radius [Equation goes here - download the original to see it.] Exercise 1 Solutions 1. Let [Equation goes here - download the original to see it.] .[Equation goes here - download the original to see it.] Therefore, we eliminate the a from [Equation goes here - download the original to see it.] From the second equation we have [Equation goes here - download the original to see it.]. Substituting this relation in the first equation, we have [Equation goes here - download the original to see it.] So [Equation goes here - download the original to see it.] Geometrically, this means [Diagram goes here - download the original to see it.] 2. We have [Equation goes here - download the original to see it.] , where a varies. Let [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] From [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] , we obtain [Equation goes here - download the original to see it.] and substituting this in the first relation, we obtain [Equation goes here - download the original to see it.] Geometrically, this means [Diagram goes here - download the original to see it.]
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Envelopes
1 Envelopes - Exercise 1A - Problems in finding envelopes
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