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a) Let P be any point on the ellipse E: x²y² a² 62 + 1. The normal at P is the line through P perpendicular to the tangent at P. α 2² (i) Show that the equation of the normal at P = M a²case bise cosio ax sin by cos 0 = (a²- 6²) sin cos a'sinecase-sin coso (ii) The midpoint, M, of two points (x₁, y₁), (x2, y₂) is given by the formula: (a cos , b sin ) is *-(5=²*+*) x1 + x2 Y1+ y2 2 2 Let G and G' be the x and y intercepts respectively of the normal at P. Show that the midpoint of GG' lies on the ellipse given by equation a²x² + b²y² = ²(a² − 6²)².

a) Let P be any point on the ellipse E: x²y² a² 62 + 1. The normal at P is the line through P perpendicular to the tangent at P. α 2² (i) Show that the equation of the normal at P = M a²case bise cosio ax sin by cos 0 = (a²- 6²) sin cos a'sinecase-sin coso (ii) The midpoint, M, of two points (x₁, y₁), (x2, y₂) is given by the formula: (a cos , b sin ) is *-(5=²*+*) x1 + x2 Y1+ y2 2 2 Let G and G' be the x and y intercepts respectively of the normal at P. Show that the midpoint of GG' lies on the ellipse given by equation a²x² + b²y² = ²(a² − 6²)².

Elementary Geometry For College Students, 7e
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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a) Let P be any point on the ellipse E: accasio beste 2² The normal at P is the line through P perpendicular to the tangent at P. а (i) Show that the equation of the normal at P = (a cos , b sin ) is ax sin by cos 0 = (a²- 6²) sin cos a² + = 1. a²sinecase-bsncoso (ii) The midpoint, M, of two points (x1, y₁), (x2, y2) is given by the formula: M = x1 + x2 Y₁+ y2 2 2 Let G and G' be the x and y intercepts respectively of the normal at P. Show that the midpoint of GG' lies on the ellipse given by equation a²x² + b²y² = = (a²- 6²)². b) Find the set of points which are the same distance from the lines x + 3y + 4 = 0 and x + 3y + 6 = 0, justifying your answer. c) Sketch the following equation which is given in Cartesian coordinates by changing to polar coordinates. (x² + y²)² = 2xy.
Transcribed Image Text:a) Let P be any point on the ellipse E: accasio beste 2² The normal at P is the line through P perpendicular to the tangent at P. а (i) Show that the equation of the normal at P = (a cos , b sin ) is ax sin by cos 0 = (a²- 6²) sin cos a² + = 1. a²sinecase-bsncoso (ii) The midpoint, M, of two points (x1, y₁), (x2, y2) is given by the formula: M = x1 + x2 Y₁+ y2 2 2 Let G and G' be the x and y intercepts respectively of the normal at P. Show that the midpoint of GG' lies on the ellipse given by equation a²x² + b²y² = = (a²- 6²)². b) Find the set of points which are the same distance from the lines x + 3y + 4 = 0 and x + 3y + 6 = 0, justifying your answer. c) Sketch the following equation which is given in Cartesian coordinates by changing to polar coordinates. (x² + y²)² = 2xy.
Question 4 Let C be the conic section with equation: 11x² + 10√3xy + y² + 4(3 − 4√3)x − 4(3√3+ 4)y − 36 = 0. 6²40c1/3)-44X1 = 300-44=256=16² a) Use the discriminant to determine what type of conic section C is. b) Determine what angle a is needed to rotate the original (x, y)-coordinate system shyperbola. anticlockwise to a new (X, Y)-coordinate system to eliminate the ïy term. B tanza In the new (X, Y)-coordinate system, we find that the equation of the conic section is -A-C. 4X² - 8X - Y² - 6Y = 9 ST c) Find the eccentricity, centre, any vertices, foci, directrices and any asymptotes of C in this coordinate system. d) Give a sketch of C including its centre, any vertices, foci, directrices and any asymptotes in this coordinate system. e) Hence give a rough sketch of C in the original (x, y)-coordinate system by rotating clockwise by a.
Transcribed Image Text:Question 4 Let C be the conic section with equation: 11x² + 10√3xy + y² + 4(3 − 4√3)x − 4(3√3+ 4)y − 36 = 0. 6²40c1/3)-44X1 = 300-44=256=16² a) Use the discriminant to determine what type of conic section C is. b) Determine what angle a is needed to rotate the original (x, y)-coordinate system shyperbola. anticlockwise to a new (X, Y)-coordinate system to eliminate the ïy term. B tanza In the new (X, Y)-coordinate system, we find that the equation of the conic section is -A-C. 4X² - 8X - Y² - 6Y = 9 ST c) Find the eccentricity, centre, any vertices, foci, directrices and any asymptotes of C in this coordinate system. d) Give a sketch of C including its centre, any vertices, foci, directrices and any asymptotes in this coordinate system. e) Hence give a rough sketch of C in the original (x, y)-coordinate system by rotating clockwise by a.
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