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2. (a) Let L be a set of points in the plane. Consider the following statement S: "If L is described by the equation y = mx + c for some m, c ER, then L is a straight line." (i) Is S true? (ii) Write down the converse of S. (iii) Is the converse of S true?

2. (a) Let L be a set of points in the plane. Consider the following statement S: "If L is described by the equation y = mx + c for some m, c ER, then L is a straight line." (i) Is S true? (ii) Write down the converse of S. (iii) Is the converse of S true?

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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2. (a) Let L be a set of points in the plane. Consider the following statement S: "If L is described by the equation y = mx + c for some m, c E R, then L is a straight line." (i) Is S true? (ii) Write down the converse of S. (iii) Is the converse of S true? Briefly explain your answers. (b) In each of the following situations find a formula describing the locus of the point P, in Cartesian coordinates, and briefly describe the shape of the locus. (It is not necessary to give detailed geometric properties of the shape such as its centre.) (i) The distance from P₁ to the point (-2,2) is equal to that from P₁ to (0,4). (ii) The distance from P₂ to the point (1,3) is twice that from P₂ to (-1,2). (iii) The distance from P3 to the line 3y = 4x is 1. (iv) The distance from P to the point (-5, 3) is twice the distance from P4 to the line with equation 4x + 3y + 5 = 0.
Transcribed Image Text:2. (a) Let L be a set of points in the plane. Consider the following statement S: "If L is described by the equation y = mx + c for some m, c E R, then L is a straight line." (i) Is S true? (ii) Write down the converse of S. (iii) Is the converse of S true? Briefly explain your answers. (b) In each of the following situations find a formula describing the locus of the point P, in Cartesian coordinates, and briefly describe the shape of the locus. (It is not necessary to give detailed geometric properties of the shape such as its centre.) (i) The distance from P₁ to the point (-2,2) is equal to that from P₁ to (0,4). (ii) The distance from P₂ to the point (1,3) is twice that from P₂ to (-1,2). (iii) The distance from P3 to the line 3y = 4x is 1. (iv) The distance from P to the point (-5, 3) is twice the distance from P4 to the line with equation 4x + 3y + 5 = 0.
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