4. Suppose that the x- and y-axes are rotated through an angle a to give new axes labelled by X and Y. You may assume that: x = X cos a - Y sin a, y = X sin a + Y cosa. (a) Write X and Y in terms of x and y, and give a geometrical explanation for why these formulas are similar to the formulas above. (b) Let C be the conic with equation: x² + y²-2ry-8√2x-8√/2y = 0. A new system of axes is introduced as above. Find a value of a such that the coefficient of XY in the new equation for C is zero. (c) Let & be an ellipse given in standard form: 1. 32 22 Find a value of a, with 0 < a < 2, with the property that the equation of & in X, Y-coordinates is: Y² X² 22 3² = 1. (d) Let C be a curve with equation px² + qy² = r, where p, q, r are all positive and Show that dard allings
4. Suppose that the x- and y-axes are rotated through an angle a to give new axes labelled by X and Y. You may assume that: x = X cos a - Y sin a, y = X sin a + Y cosa. (a) Write X and Y in terms of x and y, and give a geometrical explanation for why these formulas are similar to the formulas above. (b) Let C be the conic with equation: x² + y²-2ry-8√2x-8√/2y = 0. A new system of axes is introduced as above. Find a value of a such that the coefficient of XY in the new equation for C is zero. (c) Let & be an ellipse given in standard form: 1. 32 22 Find a value of a, with 0 < a < 2, with the property that the equation of & in X, Y-coordinates is: Y² X² 22 3² = 1. (d) Let C be a curve with equation px² + qy² = r, where p, q, r are all positive and Show that dard allings
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Transcribed Image Text:4. Suppose that the x- and y-axes are rotated through an angle o to give new axes labelled
by X and Y. You may assume that:
x = X cos a - Y sin a,
y X sin a + Y cos a.
(a) Write X and Y in terms of x and y, and give a geometrical explanation for why
these formulas are similar to the formulas above.
(b) Let C be the conic with equation:
x² + y² - 2xy-8√√2x - 8√/2y = 0.
A new system of axes is introduced as above. Find a value of a such that the
coefficient of XY in the new equation for C is zero.
(c) Let & be an ellipse given in standard form:
x²y²
+ = 1.
3² 22
Find a value of a, with 0 < a < 2, with the property that the equation of & in
X, Y-coordinates is:
X² Y²
+ = 1.
22 3²
1.
(d) Let C be a curve with equation pr² + qy² = r, where p, q, r are all positive and
p<q. Show that C is a standard ellipse.
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