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Question
I have a question, could please answer how to solve

a) the x-intercept A and y-intercept B of the line.

b) the coordinates of the point P so that lin AP is perpendicular to
the line and the magnitude of of the line AP = 1. (There are two answers.)

of the equation (3x)-(4y)=12

Answer
Rewrite the equation for the line as

y = -3 + 3x/4.

The x-intercept (y = 0) is  0 = -3 + 3x/4 or x = 4 = A

y-intercept (x = 0)  is y = -3 = B

A line perpindicular the this line has a slope that is the negatvie inverse of the line's slope. For P =(Px, Py) and the line going thru A and B,

slope = -4/3 = (Py - 0)/(Px - 4) or Py = (-4/3)(Px - 4).

The magnitude of the line = 1 means (Px^2 +Py^2)^1/2, substituting the expression for Py give Px will give you one of your solutions. The other solution is for Py -> -Py.

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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