Question 389043
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Hi
Note: the standard slope-intercept form for an equation of a line is y = mx + b  
where m is the slope and b the y-intercept.  
line through A(3,1) perpendicular to x - 4y = 8 OR y = (1/4)x - 2  
Perpendicular lines have slopes that are negative reciprocals of one another
New line y = -4x + b using ordered pair Pt(3,1) to solve for b
         1 = -12 + b     b = 13
         y = -4x + 13   P(13/4,0) and Q(0,13)on this line
Calculate the ratio PA : AQ
PA Distance would be: P(13/4,0)and A(3,1)
{{{sqrt((x[2] - x[1])^2 + (y[2] - y[1])^2)}}}
PA = {{{sqrt((-.25)^2 + 1^2) = sqrt(1.0626)= 1.031  }}}
AQ Distance would be: A(3,1)and Q(0,13)
AQ =  {{{sqrt((-3)^2 + (12)^2) = sqrt(153) = 12.369 }}}
PA:AQ would be   1.031:12.369
{{{drawing(300,300, -6, 16, -6, 16, grid(1),
circle(13/4, 0,0.4),
locate(13/4,0,"P"),
circle(0, 13,0.4),
locate(0,13,"Q"),
circle(3, 1,0.4),
locate(3,1,"A"),
graph( 300, 300,-6,16,-6,16, .25x -2, -4x + 13)) }}}