PP Section 11.1

# 1 2 1 2 x x y y m 2 3 4 6 3 1 8 2 3 2 3 8 x y

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1 2 1 2 x x y y m - - = 2 3 4 6 3 1 8 2 = - - = - - - = ) 3 ( 2 3 8 - = - x y

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Definition The angle of inclination of a line y = mx + b is the positive angle θ from the positive x -axis to the line. θ Fact: m = tan(θ) π θ < 0 : Fact
Example Find the angle of inclination of the line - 4x + 2y = 9. 2 9 2 9 4 2 + = + = x y x y ) tan( 2 θ = = m ( 29 4 . 63 or 1071 . 1 2 tan 1 - Since the answer obtained with the calculator is in [0, π) or [0, 180°), then θ = 63.4°.

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Example Find the angle of inclination of the line 7x + 5y = 1. 5 1 5 7 1 7 5 + - = + - = x y x y 5 7 - = m 54.5 - or 95055 . 0 5 7 tan 1 - - - The answer obtained with the calculator is NOT in [0, π) or [0, 180°), so θ = 180° – 54.5° = 125.5°
Distance from a Point to a Line The distance d from a point (a, c) to a line with equation y = mx + b is 2 1 m c b ma d + - + =

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Example Find the distance from (-1, 3) to 2x + y =7. We need the equation in slope-intercept form, so y = -2x + 7, 2 1 m c b ma d + - + = 2 ) 2 ( 1 3 7 ) 1 ( 2 - + - + - - = 5 6 =
Example P From the point P(5, 0) a line is drawn tangent to the circle, as shown.

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