   Chapter 11.4, Problem 70E ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

#### Solutions

Chapter
Section ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

# SKILLS PlusDeterminant Form for the equation of a Line(a) Use the result of Exercise 69(a) to show that the equation of the line containing the points ( x 1 , y 1 ) and ( x 2 , y 2 ) is | x y 1 x 1 y 1 1 x 2 y 2 1 | = 0 (b) Use the result of part (a) to find an equation for the line containing the points ( 20 , 50 ) and ( − 10 , 25 ) .Collinear Points and Determinants(a) If three points lie on a line, what is the area of the “triangle” that they determine? Use the answer to this question, together with the determinant formula for the area of a triangle, to explain why the points ( a 1 , b 1 ) , ( a 2 , b 2 ) and ( a 3 , b 3 ) are collinear if and only if | a 1 b 1 1 a 2 b 2 1 a 3 b 3 1 | = 0 (b) Use a determinant to check whether each set of points is collinear. Graph them to verify your answer.(i) ( − 6 , 4 ) , ( 2 , 10 ) , ( 6 , 13 ) (ii) ( − 5 , 10 ) , ( 2 , 6 ) , ( 15 , − 2 )

To determine

(a)

To prove:

The statement “the equation of the line containing the points (x1,y1) and (x2,y2) is |xy1x1y11x2y21|=0”.

Explanation

Approach:

The equation of the line containing the points (x1,y1) and (x2,y2) is given by,

yy1=y2y1x2x1(xx1).

Calculations:

Consider the points points (x1,y1) and (x2,y2) and the determinant equation |xy1x1y11x2y21|=0.

Perform the transformation R1R1R2.

|xx1yy10x1y11x2y21|=0

Perform the transformation R2R2R3.

|xx1yy10x1x2y1y20x2y21|=0

Expand the determinant equation as shown below

To determine

(b)

To find:

The equation of the line passing through the points (20,50) and (10,25).

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