Coordinate Geometry

Question

The line through A(3,1) perpendicular to x-4y =8 meets the x-axis at P and the y-axis at Q.Calculate the ratio PA:PQ.

 

Ans:  1:12 (Show working please)

Answer 1

First, put the equation x - 4y = -8 in slope-intercept form:  

 

            -4y = -x - 8    So, y = (1/4)x + 2

 

              The given line has slope 1/4.  Since the slopes of                                     perpendicular lines are negative reciprocals of each other,                 any line perpendicular to the given line must have slope -4.

 

An equation of the line through (3, 1) with slope -4 is:

 

     y - 1 = -4(x - 3)

 

     Simplify to get y = -4x + 13

 

     x-intercept:  (set y = 0)   P = (3.25, 0)

     y-intercept:  (set x = 0)   Q = (0, 13)

 

     PA = √[(3.25 - 3)² + (0 - 1)²] = √(17/16)

     QA = √[(0 - 3)² + (13 - 1)²] = √153

 

     PA/QA = √(17/2448) = √(1/144) = 1/12